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MATHEMATICS 


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ELEMENTS OF 
THE THEORY" OF-ENTEGERS 


ELEMENTS OF THE 


MeO Or NT EGERS 


BY 


JOSEPH BOWDEN, Px.D. 


PROFESSOR OF MATHEMATICS IN ADEL PHI COLLEGE, BROOKLYN » NEW YOR K 


Het Work: 
THE MACMILLAN COMPANY 
LONDON : MACMILLAN & CO., LTD, 


noe 3 


CoPpYRIGHT, 1903 
By THE MACMILLAN COMPANY 


Set up and electrotyped October, 1903 


PRESS OF 
THe New ERA PRINTING COMPARY, 
LANCASTER, PA, 


va 
teen 
(v7 


SELEY oe Ty 
TT) 


2. oe . 


PREACH) 


The present volume contains the first two parts of 
a work which the author hopes to finish, on the Ele- 
ments of the Theory of Numbers, the remaining three 
parts being devoted to Rational Numbers, Real 
Numbers, and Complex Numbers, respectivly. 

The book has sprung from a desire to put the ele- 
mentary theory of numbers in a logical form, starting 
from the three fundamental ideas of number, equality 
and sum, with their axioms, building up a system of 
theorems on these fundamental ideas, and:then, by as 
natural a process as possible, introducing the derived 
ideas of greater, less, difference, integer, product, 
Juotient, and so forth. 


JosEPH BOWDEN. 
BROOKLYN, NEW YorRK, 


January 3, 1903. 





UT ABER OP REO NRE ING S: 


EARS 


PRIMARY NUMBERS. 


GHALLE RSI: 


FUNDAMENTAL IDEAS. AXIOMS. DEFINITIONS. 


NuMBER. EQUALITY. ADDITION. 
ARTICLE, 


1. Number . 

rae LORY. 

5. Equality 

6, 10. Axiom 

11. Theorem : : : : - : : 
12. Statements, categorical, hypothetical, and disjunctiv . 
37. Sum : 7 : 

39. Univalent symbols . 

41. Commutativ law for addition 

42. Operation of addition 

54. Associativ law for addition 

60. Parts . 

64. Unity . : : : 

67. Natural series of numbers 

70. Correspondence 

71. Counting. ; : : ; : ° ° 
74. Addition table A : : : : C . 


CHAPTER TIE 


SUBTRACTION. 
GREATER. Less. DIFFERENCE. 


75. Greater and less ‘ - : ° “ 
77. Difference ; : ; : : : : ; 
Vii 


a2 
a8, 


viil 


TABLE. OF CONTENTS. 


ARTICLE. 


84. 
527, 
130. 
137. 
144. 
145. 
146. 
147. 
150. 


15a: 
161. 
168. 
186. 
203. 
207. 
220. 
249. 
PAM f 
259. 
261. 
266. 
287. 
295. 


2096. 
207. 


Implicit and explicit definitions. Derived ideas 
Necessary and sufficient conditions 

Syllogisms : : : 

Infinit number. Finite number 

Numbers orderd 

Operation of subtraction 

Subtraction table 

Inverse of a given operation 

Synthetic and analytic operations 


PAR BP: Ib 
INTEGERS. 


CHAPTER: III. 


Positiv INTEGERS. ZERO. NEGATIV INTEGERS. 


Integers: ; : : 

Opposit of a given integer 

Equality . 

Greater. Less 

Sum : ; : A : . 

Commutativ law for addition of integers 

Difference : 5 ° ° 

Associativ law for addition of integers ‘ 
Associativ law for addition and subtraction of integers 
Commutativ law for addition and subtraction of integers 
Cancellation 

Transposition . : : 

Even integers. Odd integers . 

Closed system . 


CHAPTER IV. 


MULTIPLICATION. 


Product, coefficient a primary number. Multiple. 
Operation of multiplication . ; ; : : 


61 
64 
66 
70 
74 
75 


al 
SI 


85 
86 
86 
88 


93 
oF 


98 
99 


vF 


TABLE OF CONTENTS. 


ARTICLE, 
310, 311. Distributiv laws for multiplication . 


313- 
326. 
375- 
376. 
380. 


392. 


393- 
400. 
402. 


404. 


409. 
TAL2. 
4106. 
478. 
490. 
504.. 
5006. 
508, 


513. 


514. 


538. 
545- 


Product, coefficient an integer . 
Rule of signs for multiplication. 
Commutativ law for multiplication 
Factors : : : 
Associativ law for multiplication 
Multiplication table . 


CHAPTER V. 


NUMERICAL VALUE. 


Numerical value 

Numerical value of a sum. 
Numerical value of a difference. 
Numerical value of a product 


CHARTER Vis 


DIVISION. 


DIVISIBILITY. FACTORS. QUOTIENT. 


Divisible by, divisor of : 
Quotient. : ; : : 5 
Multivalent and indeterminate symbols 
Distributiv law for division and addition 
Rule of signs for division . 

Axiom of Archimedes 

Operation of division 


Division table . f : : : : : 
Exact division. Approximate division. Lower and upper 
quotients 
Remainder ; ‘ 
CHAPTERS VLE 
FACTORS. 
Prime integers. Composit integers . ‘ 
Table of prime integers. R A A 


123 
124 
126 
126 


128 
129 
130 
144 
147 
151 
152 
154 


157 
157 


167 
169 


x 


TABLE OF CONTENTS. 


ARTICLE. 


550. 
B51: 
555: 


572. 
573: 
574- 
594- 
598, 


602. 
607. 
637. 
646. 
662. 


674. 
676. 
680. 
681. 
691. 


701, 


704. 


708. 


731. 
766. 


767. 


Resolution of a composit integer into prime factors. - 
Common factors. : : : : ; ; . 
Relativ primeness 


CHAPTER TY itl 


GREATEST COMMON FACTOR. 


Greatest common factor 

Operation of finding greatest common factor 

Commutativ law for greatest common factor 

Algorithm of Euclid 5 : 5 

599. Distributiv laws for multiplication and greatest com- 
mon factor ; - : : 

Distributiv law for division and greatest common factor 

Mathematical induction : : : : : 

Representation of numbers by products of prime factors 

Associativ law for greatest common factor . 

Integers prime to each other two by two 


CHAPTER IX. 
LEAST COMMON MULTIPLE. 


Common multiple 

Least common multiple : : 

Operation of finding least common multiple 

Commutativ law for least common multiple 

Connection between least common multiple and greatest 
common factor : : : : : 

702. Distributiv laws for multiplication and least common 
multiple ‘ : . : : 

Distributiv law for division and least common multiple 

Associativ law for least common multiple . 


CHAPTER: X, 


CONGRUENCE. 
Congruence. Modulus : - : : ; 
Periodicity of the integers with respect to a given modulus. 
Cyclical arrangement. : , 


PAGE, 


173 
174 
174 


177 


181 


188 
201 
207 


JEVAMECIP db 
PRIMARY NUMBERS. 


ClebeUe eas IL. 


FUNDAMENTAL IDEAS. AXIOMS. DEFINITIONS. NUM- 
BER. EQUALITY. ADDITION, 


1, The concept of number, in its simplest and origi- 
nal sense, is a fundamental concept. It is incapable 
of definition, that is, it cannot be exprest in terms 
of ideas simpler than itself. 

The concept is aroused in the mind by the consid- 
eration of groups of objects, material or mental (°), like 
or unlike. To every group of objects(*) belongs a 
number, the number of objects in the group. 

2. To every single object, also, a number belongs, 
the number one. 

3. Altho the word group refers properly to a col- 
lection of objects, we will, for the sake of brevity, use 
it to represent either a single object or many objects. 


(1) See Locke, ‘‘An Essay Concerning Human Understanding,”’’ 
1690, Bk. II., p. 84; Schubert, ‘‘ Encyklopidie der Mathematischen 
Wissenschaften,’’ 1898, Vol. I., p. 1. 

(2) See Lefevre, ‘‘ Number and its algebra,’’ 1896, p. 19 ; Stieltjes, 
Annales de la Faculté des Sciences de Toulouse, 1890, Vol. 1V., p. 1. 


I 


2, PRIMARY NUMBERS. 


4, Notation. Each number, like any other idea, 
mav be given a name('). Thus we hav the numbers 
one, two, nve. “ Numbers may also be represented by 
symbols, as 2, 7, 9. 

Frequently we shall hav occasion to make a state- 
ment that is true of all numbers. In this case let- 
ters (*) of the alphabet will be used, each to represent 
any number we choose to consider. If a particular 
letter occurs more than once in a statement, it will each 
time stand for the same number. 

5. Definition. The numbers of objects in two 
groups may, or may not, be the same. In this idea 
of sameness between two numbers we hav another 
simple, or fundamental concept. To it we giv the 
special name equality, to distinguish the sameness 
between numbers from the sameness between other 
things. 

If the numbers of objects in two groups ar the 
same, they ar said to be equal(*). Representing these 
numbers by a and @ respectivly and using the sign 
= for the words “is equal to(*),” we may write this 
statement of equality a= Jd, 





(1) See Ball, ‘‘A Short Account of the History of Mathematics,’’ 
1893, p. 123: 

(2) Letters were sometimes used in this way to represent numbers by 
the ancient Greeks. The first modern European mathematician to use 
letters for such a purpose was Jordanus Nemorarius, ‘‘ Algorithmus 
Demonstratus,’’ 1534. See Schubert, ‘‘ Encyklopiadie,’”’ p. 5. 

(3) See Tannery, ‘‘ Legons d’ Arithmétique,’’ 1900, p. 3. 

(4) The sign = seems to hav been used for the idea of equality first 
by Record, ‘‘ Whetstone of Witte,’’ 1557. 


FUNDAMENTAL IDEAS. 3 


The expression @= 0 is calld an equality and is 
read ‘a is equal to 6” or “aequals J.” It is merely 
an.abbreviation for the statement: The number of 
objects in the first of the two groups under considera- 
tion is the same as the number in the second. 

The number on the left of the = sign is calld the 
left or first member of the equality, the other number 
the right or second member. 

If a is not equal to 4, we write a+ 6, This expres- 
sion is calld an inequality. 

6. Axiom. Axy number 1s equal to itself (’). 

In symbols: @= a. 

A longer statement of the same principle is: The 
number of objects in a group ts the same as the number of 
De ia eT OU ee OtmeKaMple,.2—2, 0-070 

The mind grants the truth of this principle as soon 
as it grasps the meaning of the words in which it is 
stated. All such principles, with a restriction to be 

given later (§ 10), ar calld axioms(’). 
~ 4%, Axiom. a=), thend= a(*). 

In words: /f the number of objects in a group ts the 
same as the number of objects in a@ second group, then 
the number of objects in the second group is the same as 
the number of oljects in the first group. 

This principle also is axiomatic. It may be calld 
the Commutativ Law for Equality. 


(1) See Stolz und Gmeiner, ‘‘ Theoretische Arithmetik,’’ 1900, p. 2. 
(?) See Jevons-Hill, ‘‘ Elements of Logic,’’ 1883, p. 112. 
(3) See Schubert, ‘‘ Encyklopiadie,’’ p. 5 ; Stolz und Gmeiner, p. 2. 


4 PRIMARY NUMBERS. 


8. Theorem. /fa+), thenb +a. 

For it follows directly from § 7 that, if da, then 
ae= 0. 

In words this says that ‘if a first number, which 
we call J, is not equal to a second number a, then 
the second number @ is not equal to the first num- 
Dero,” 

If we call the first number a instead of 6, and call 
the second number 4, this statement will be: If 
@=+'0, then O= a. 

9. Definition. In the last two articles we hav stated 
two truths, each of which is as axiomatic, in the sense 
of § 6, as the other and each of which follows as a 
logical consequence from the other. It is customary 
in such a case not to call both of the truths axioms, 
but to call one an axiom and the other a theorem. 

10. Definition. In general, if for any subject ther 
is a group of several axiomatic principles, of which 
one or more is a consequence of one or more of the 
others, it is customary to choose from this group a 
smaller group, such that no one of the principles in 
the smaller group is provable from one or more of 
the others in that group. 

The axioms of the smaller group ar then said to 
be independent (*) of each other. 





(1) See Hilbert, ‘‘ The Foundations of Geometry,’’ translated from 
the German by Townsend, 1902, p. 1; Moore, Zransactions of the 
American Mathematical Society, 1902, Vol. III., p. 142; Hunting- 
ton, Zransactions of the American Mathematical Society, 1902, Vol. 
LIT., pe 264. 


FUNDAMENTAL IDEAS. 5 


The chosen group must also be such that all the 
other principles in the given group and all other 
known principles in the subject under consideration 
shall be logical consequences of the axioms in the 
smaller group. | 

The axioms of the smaller group ar then said to 
be sufficient for the subject under consideration. 

Usually the choice of the smaller group can be 
made in several ways. An endeavor should, however, 
be made to hav the axioms which are chosen as simple 
in statement and as few in number as possible. 

If ther is no choice in these respects, or if these 
two requirements conflict with each other, the choice 
is made arbitrarily. 

We then hav what is calld an independent and 
sufficient group of axioms for the subject under con- 
sideration. 

Whether the axioms that will be set up in this 
chapter constitute an independent and sufficient group 
is left for future discussion. 

That these axioms ar consistent, or compatible (’), 
with each other, that is, not contradictory, is evident 
from the fact that they ar all true of the system of 
numbers. 

11. Definition. A theorem is a principle whose truth 
follows as a logical consequence from the axioms 
chosen. 


(1) See Hilbert, Bulletin of the American Mathematical Society, 
1902, Vol VIL. p.2447. 


6 PRIMARY NUMBERS. 


Thus of the given group of principles mentiond in 
§ ro all those except the ones chosen for the inde- 
pendent and sufficient group of axioms ar theorems. 

12. We will insert here a few principles of logic that 
will be useful in our future work. 

Definition. Statements (’) ar either unconditional or 
conditional, the former being calld also categorical 
statements. : 

A categorical statement may be represented sym- 
bolically by the form “A is B.”’ 

Conditional statements ar of two kinds, hypothetical 
and disjunctiv (7). 

A. hypothetical statement may be represented by 
the. formate 18a5. thengC sm). al taconsisis on 
two parts, a hypothesis and a conclusion. The hy- 
pothesis is the clause introduced by the word “if,” 
the conclusion the clause introduced by the word 
ethene: 

A disjunctiv statement may be represented by the 
form “ Either A is B, or Cis D(*).” 

13. Definition. The contradictory (*) of a statement 
is a statement to the effect that the given statement is 
false. 

Thus the contradictory of the categorical statement 
(Acts 2 1s the statement is mou... 


(1) See Jevons-Hill, pp. 66, 67. (2) See Jevons-Hill, p. 150. 

(3) See Jevons-Hill, p. 156. 

(+) See Jevons-Hill, p. 83 ; Wentworth, ‘‘Plane and Solid Geom- 
etry,’’ I90I, p. 5. 


FUNDAMENTAL IDEAS. 7 


The contradictory of the hypothetical statement 
lieicin Bothen Cis weis thestatement ‘olf is 2, 
then’ Gis not JD.” 

The contradictory of the disjunctiv statement 
meratiermdis x or Gis > is“ Neither Ais -bnor 
C is D,” which is equivalent to “ A is not B and C is 
stale 7OEM 

14. Axiom. Evidently a statement and tts contradic- 
tory cannot both be true. 

15. Axiom, Zhe contradictory of the nD ie of 
a given statement ts the original statement. 

Thus the contradictory of the contradictory of the 
statement “A is £” is the statement “4 is not not 
eerOret A iS i 

16. Definition. The opposit of a given conditional 
statement, hypothetical or disjunctiv, is a conditional 
statement of the same kind, whose parts ar respec- 
tivly the contradictories of the parts of the given 
statement. 

Thus the opposit of the hypothetical statement 
elie ism mulcmaOetss/), sisathe statement, “lte4 15 
no.-6, then, Cis not-27.”’ 

The opposit of the disjunctiv statement “ Either A 
ise ome ice 4 isathe statement. Hither 41s not 6, 
otc is not)” 

17. Theorem. The opposit of the opposit of a given 
conditional statement is the original statement. 

This follows immediately from §§ 15, 16. 


8 PRIMARY NUMBERS. 


18. Definition. Statements ar simple or compound (’). 
A simple categorical statement is of the form “4 
is B,”’ 

A compound categorical statement consists of two or 
more simple categorical statements, usually connected 
by the word #Sand.” Thus#'4ns Sandie) is 4 
compound categorical statement. 

19. Axiom. The contradictory of the compound cate- 
gorical statement “A ts B and Cis D” is the disjunctiv 
statement “Lather A is not B, or C 1s not D.” 

20. Definition. A simple conditional statement is of 
one of the two forms “If A is 4, then C is D”’ and 
‘(Ritherid 1s28 <orCus a 

A compound conditional statement, hypothetical or 
disjunctiv, is a conditional statement of the same kind, 
of which either the first or the second part is com- 
pound. Thus the statements “If 4 is 6 and Cis D, 
then £ is F,” “If A is B, then C is D and Z£ is F,” and 
‘CfA Ms8 Brandis 2 then aise hare te eel 
all compound hypothetical statements ; the statement 
‘Either Ais Gand Cis D, or 4s ange asi 18 
a compound disjunctiv statement. 

21. Definition. The converse (*) of a given hypothet- 
ical statement is a hypothetical statement whose hy- 
pothesis is the conclusion of the given statement and 
whose conclusion is the hypothesis of the given state- 





(1) See Swinton, ‘‘ New English Grammar,’’ 1881, p. 211. 
(2) See Wentworth, p. 5. 


FUNDAMENTAL IDEAS. 9 


ment. Thus the converse of “If.A is B, then Cis D” 
ieoelie(aises) thén=A4istpe’ 

To distinguish a given hypothetical statement from 
its converse, the given statement is calld the direct 
statement. 

22. Axiom. The converse of the converse of a given 
hypothetical statement is the original statement. 

23. Axiom. The converse of the opposit of a given 
hypothetical statement 1s the oppostt of the converse of 
_ that statement. 

24, Axiom. Jf a hypothetical statement 1s true, the 
opposit of its converse 1s true. 

Mhtissiethie statement.“ lf Aus tbathen Cus 07 as 
true, so also is the statement ‘If C is not D, then A 
Ismnot Dx 

25. Theorem. Jf the opposit of a given hypothetical 
statement ts true, its converse ts true, 

This follows directly from the preceding. 

Thus, if the statement ‘If A is not 4, then C is not 
eis ttuesby.si24 we hav “liCuis DO then A is 2.5 

26. Theorem. Jf the converse of a given hypothetical 
statement ts true, its opposit is true. 

Thus, if the. statement “If Cis, D, then 4 is B” is 
true, so also is the statement “If A is not 4, then C is 
not D.” 

27. The theorem of § 8 follows from the axiom of 
§ 7 by the application of the logical rule of § 24. 

28. Corresponding to every hypothetical statement 
ther ar three others, its opposit, its converse, and the 


IO PRIMARY NUMBERS. 


opposit of its converse. We will number these as 
follows : 

1. Given statement. 

2. Its opposit. 

3. .ltseconverse: 

4. The opposit of its converse. 

These four statements may be groupt in pairs, I 
and 4 together and 2 and 3 together, so that, if either 
one of a pair is true, the other one of that pair is also 
true. We may say that I and 4 ar equivalent to 
each other and 2 and 3 equivalent to each other. 


29. If a hypothetical statement is true, its opposit - 


is not necessarily true. Thus, if the statement “ If 
Ais 5, then’ Cis: DV eis  truessominay.aiso spesthe 
statement,“ A is: note athenaCeisw) a that aise tic 
statement) “If Ais not*SAthen CMsinotew vamay not 
be true. Similarly, if a hypothetical statement is true, 
its converse is not necessarily true. 

30. Any hypothetical statement of the form “If 
A is L, then Cis D” and its equivalent “If C is not 
D, then A is not &” ar equivalent to the disjunctiv 
statement ‘‘ Either A is not B, or Cis D.” 

Similarly any disjunctiv statement of the form 
“Bither A is B, or C is D”’ istequivalent to either of 
the hypothetical statements “If A is not B, then C is 
Di ands li Cuismote/ there is eee 

31. By replacing the disjunctiv statement “ Either 
A is b, or C is D”’ by its equivalent hypothetical 
statement, it is plain that, if this disjunctiv statement 


a 


FUNDAMENTAL IDEAS. tet 


feecruc eiiee statement. iBoth 24 iso ands G is7iv/ 
may also be true. 

If, however, the opposit disjunctiv statement “ Either 
famic T1Ote, Or C is- not 021s true, then only one 
Oigticestatcients {Ais Grand “Gris Ds” canbe 
sce wevve Gan in this*casessay*that-“lf A is 2B; 
Pacnm(eisenot /J< -andetiate alin Gis /).“theny Asis 
not B.”’ 

When the given disjunctiv statement and its oppo- 
sit ar both true, we can say that one, and only one, 
Omiuesstatcments= eis, 5b. sande: Cis) is true; 

32. Definition. A partial converse of a given com- 
pound hypothetical statement is a hypothetical state- 
ment whose hypothesis contains one of the clauses 
of the conclusion of the given hypothetical statement 
and whose conclusion contains one of the clauses of 
the hypothesis of the given statement. 

Thus a partial converse of the statement “If A is B 
gdec ise) tnene/ise// and Gusi/7 “1s the statenient 
Meelueeistand: Ase) thens@ is 2 and-Gis Au 


33. We will now resume our discussion of the theory 
of numbers with the remark that the equality a= 0 is 
an illustration of one of the chief causes for the great 
growth of the science of mathematics. This is that 
mathematicians have gradually developt a system of 
symbols by means of which the most complex mathe- 
matical ideas may be represented in a clear, compact, 
unambiguous form, so that the meaning is apparent at 
a glance. 


12 PRIMARY NUMBERS. 


34. Axiom. /f a=dand b=c, then a=c(’'). 

In words: Jf the number of objects in a group is the 
same as the number of objects in a second group and the 
latter number 1s the same as the number in a third 
group, then the number in the first group ts the same 
as the number in the third group. 

Stated in this form the truth of this principle is self- 
evident. 

This axiom is usually stated: Jf two numbers ar 
equal to the same number, they ar equal to each 
other. 

35. Theorem. /f/a=0 and b +c, then a+ c(’). 

For, by § 24, it follows from the preceding axiom 


that, if a +c, then either.a + 6 or d+. § 19. 
This is equivalent to the statement : 7 
If a=éand a+c, then 6+. § 30. 
Or, if 6=aanda+c, then d+ c. $7. 


From this, interchanging the letters a and 4, we get 
the theorem. | 

36. Theorem. Generalization of § 34. Jf we hav 
a series of equalities in which the right member of each 
equality, except the last, 1s also the left member of the 
succeeding CQUualiy, Q5 0 =O 6G, pe 10 ee 
or, for the sake of brevity, a=b=c=d=ce=::., 
(the dots --- meaning “and so forth,” that is, that 
ther may be any number of equalities in the series), 


, 


(1) See Schubert, ‘‘ Encyklopadie,’? p. 6; Stolz und Gmeiner. 


ee 
(2) See Jevons-Hill, pp. 108, 109 ; Stolz und Gmeiner, p. 2. 


FUNDAMENTAL IDEAS. 13 


then any number in the series ts equal to any other num- 
ber in the series('). 

If the two numbers to be shown equal ar next each 
other in the series, ther is nothing to prove; they ar 
given equal. If ther is one number between them, 
as is the case with 4 and d, they ar equal by the 
axiom, § 34. 3 

Consider J and e, which hav two numbers between 
them. By the axiom 6=d. Then, since we hav 
b=d=e, by applying the axiom again, we see that 
ie 

So, if ther ar 4 numbers between the two num- 
bers required to be proved equal, by applying the 
axiom once, one of the intermediate numbers disap- 

pears ; by applying it & times all ar removed. 
8%. Definition. If we join two groups of objects 
so as to form a single group, the number of objects 
in the group thus formd is calld the sum(’*) of the 
numbers of objects in the two original groups. 

The latter numbers are calld the elements of the 
sum, 

If the elements of a sum ar a and 4, we will repre- 
sent the sum itself by the symbol a+ 4(°). This is 
read “a plus 0,” the sign + standing for the word 
Solus. 

(1) See Stolz und Gmeiner, p. 2. (*) See Tannery, p. 5. 

(3) The sign + was used by Widman in his ‘‘ Mercantile Arithmetic,’® 
1489. It has been conjectured that the sign was originally a warehouse 


mark. The first use of this sign to indicate a sum was in Stifel’s 
‘s Arithmetica Integra,’’ 1544. 


14 PRIMARY NUMBERS. 


38. Theorem. // c zs the sum of aand b, thenc=a+, 

For c and a + d each represent the number of ob- 
jects in the group obtaind by joining two groups of 
a and 6 objects respectivly. Sire 

Hence, since the number of objects in any group is 
the same as the number of objects in that group (§ 6), 
cis the same as a+ 0, orc=a-44J, § 5. 

39. Definition. An expression or symbol that has 
only one meaning is calld determinate, univalent, or 
‘unambiguous. 

40. Axiom. Zhe suma-+ 6 of two given numbers ts 
untvatent. 

It is evident that the number of objects in the group 
obtaind by making a single group of two given 
groups of objects is determinate. 

41, Axiom. a+ 6=6+4a(’'). 

For example,6+7=74 6. 

The expressions a+ 6 and 6+ a are two symbols 
representing the same idea, the number of objects in 
the group obtaind by joining two groups of @ and d 
objects respectivly. The numbers @ and 6 hav the 
same weight in determining the new number, their sum ; 
they ar merely written in different order in the two 
expressions a+ 6 and 0 + a. 

This axiom is calld the Commutativ(*) Law for 
Addition. 


(1) This axiom seems to hav been noticed first by Gregory, ‘‘Ona 
Difficulty in the Theory of Algebra,’’ 1840. 

(2) The word commutativ as a mathematical term was first used by 
Servois, Gergonne’s Annales, 1814, Vol. V., p. 98. 


FUNDAMENTAL IDEAS. 15 


42. Definition. In the idea of joining two groups 
of objects to form a single group, we hav a fundamental 
idea. This idea may fittingly be calld an operation. 
For it is the concept of an action, the joining of two 
groups of objects, by either a physical or a mental 
effort. To distinguish it from other operations we 
will call it the operation of addition (’). 

The sum a+ of two numbers is, then, obtaind by an 
operation. The sum may be calld the result of the oper- 
ation ; the numbers a and 6 the summands, or operands, 
the things operated on; and the sign + may be regarded 
as a symbol to indicate what operation is performd. 

The operation of addition may also be considerd 
from a slightly different standpoint. We may suppose 
that the number a is first given and that writing + 0 
after a indicates that we add 4 to a. In this sense @ 
is the operand, passiv (*) element, or augend (°), and 4 
the operator, activ element, or increment (*). 

With this understanding the commutativ law states 
that zz the operation of addition the augend and the in- 
crement may be interchanged. 

43, Axiom. /fa=)d, thna+c+ 6. 

This may be seen by considering three groups of a, 
6, and ¢ objects respectivly, the numbers of objects in 
the first two being the same. 





(1) See Schubert, ‘‘ Encyklopadie,’’ p. 6. 

(2) See Schréder, ‘‘ Abriss der Arithmetik und Algebra,’’ 1874. 

(3) See Schubert, ‘‘ Arithmetik und Algebra,’’ 1899, p. 17. 

(4) See Schubert, ‘‘ Mathematical Essays and Recreations,’’ trans- 
lated from the German by McCormack, 1898, p. Io. 


16 PRIMARY NUMBERS. 


This axiom might be stated: /fa= J, then a ts not 
equal to any sum of which b ts the activ element. 

44, Theorem. /fa=c+), thenax 8. 

This follows immediately from § 43 by the applica- 
tion of the rule of § 24. 

45. Axiom. Jf a + J, ther 1s some number c, such 
that eitthera=c+borc+a=QB, 

46. Axiom. /fa=6, thna+c=6b+4¢. 

This axiom might be stated: //f for the passiv ele- 
ment of a sum an equal number is substituted, the sum 
1s unchanged. 

47. Axiom. /fa+0,thna+tc+tb4e. 

48. Theorem. /fa+tc=b4c¢, thna=d. 

This comes immediately from § 47 by the rule of 
§ 24. 

This theorem might be stated: // the activ elements 
of two equal sums ar the same, their passtv elements ar 
equal, 

49, Theorem. /f a=30, thenc+a=c+6,; and 
conversely, ffcta=c+b, thna=od6, 

For the direct theorem, 


we hav cta=ate. § 41. 
Also, sincea=6, ate=be. § 46. 
Again 6b+c=c4+6. § 41. 
Therfor cta=c4+ § 36. 


For the converse theorem, 


FUNDAMENTAL IDEAS, iy 


we hav ate=cta .- § 41. 
and, by hypothesis, 


cta=c4+.. 
Also ct b=b+ ce, § Al. 
Hence atetr=0-+6. Sas) 
Therfor a= b, § 48. 


This theorem might be stated: Jf for the activ ele- 
ment of a suman equal number ts substituted, the sum ts 
unchanged; and conversely, if the passiv elements of two 
equal sums ar the same, their activ elements ar equal. 

50. Theorem. /f a=0 and c=d, then atc= 
6+ d('); and conversely, if a+c=b+dandc=d, 
thna=b; ffat+ec=b4+danda=b), thenc=d. 

For the direct theorem, 


since @ = 0, a@Qte=b+a | § 46. 
BINnce ¢ =a; 6+c=b6b44d. § 40. 
Therfor ate=b4d. § 34. 


This theorem is usually briefly stated: Aguals added 
to equals giv equals. 

For the first part of the converse theorem, 
we hav, by hypothesis, 


atc=64+4d. 


(1) See Stolz und Gmeiner, p. 38. 
3. / 


18 PRIMARY NUMBERS. 


Since c = d, btc=b4d. § 49. 
Hence ateo=b+e. §§7, 34. 
Therfor a= b, § 48. 


Similarly the second part of the converse theorem 
may be proved. 

It will be noticed that the path taken for the proof 
of the converse theorem is exactly the reverse of that 
taken for the proof of the direct theorem. Such is 
frequently the case. 

This theorem might also be stated: /f for the ele- 
ments of a sum equal numbers ar substituted, the sum 
7s unchanged; and conversely, if either the activ or the 
passiv elements of two equal sums ar equal, the other 
elements ar equal, 

51. Definition. Successiv Addition. After adding 
two numbers we may add their sum to a third, that 
sum to a fourth number, and so on. In order to in- 
dicate in what order the additions ar performd we will 
use parentheses (), [ ], or { }. The latter two kinds 
of parentheses ar also calld respectivly brackets and 
braces. Thus we may hav (a + 0) + ¢, the parenthe- 
sis enclosing @+ 4 signifying that the sum @ + 0 is 
formd first and then c is added to that sum (’). 

Similarly we may hav ((@2 + 6)+c)+ d,a+(d+ 0), ~ 
[a+ (b+0)]+4,a+ {((64+0)4d}, (a+0)+(c+4), 
etc. 

All such expressions ar univalent. 


(1) Parentheses were first used by Girard, 1629. 


FUNDAMENTAL IDEAS, I9 


For addition is univalent. § 40. 

Hence every expression obtaind by repeated addi- 
tion is univalent. 

Such expressions ar calld complex sums. The num- 
bers a, 0, c,---, from which a complex sum is formd, 
ar calld its elements. 

For the sake of distinction a sum that has only two 
elements is calld a simple sum. 

52. Theorem. /f for one or more elements of a com- 
plex sum equal numbers ar substituted, the complex sum 
ws unchanged’). 

This follows from §§ 46, 49, 50. 

For in building a complex sum we add only two 
numbers at a time. 

53. Definition, The standard sum of a series of 
numbers in a fixt order is the sum obtaind by adding 
the second to the first, the third to that sum, the fourth 
to that, and so on. 

For example, a+ 6, (a+ 6)+c, (@+ 6)+c¢)4+4d, 
(a+ 46)+c)+ ---+72)+ ar standard sums; wheras 
(a+ 6)+(c+d)and at [(6+e)+(¢+(e+/))] 
ar not. 

54, Axiom. (2+ 0)+c=a+4(b+4+c)(°). 

This principle becomes evident if we consider three 
eroups of a, 4, c objects respectivly. In whichever 
of the two indicated ways the groups ar joind, the 
final group is the same. 


(1) See Stolz und Gmeiner, p. 38. 
(2) This axiom seems to hav been noticed first by Gregory. 


20 PRIMARY NUMBERS. 


This axiom is calld the Associativ (") Law for Addi- 
tion. 

By this law, in connection with the commutativ 
law and §§ 46, 49, we may prove the following series 
of equalities: (2+ 6)+c=a+(6+4+c)=a+4 (c+ 3) 
=(atc)+6=64(a4c)=(64+a)+e=b4(c4+2) 
=(6+c)+a=c+ (a+ d)=(c+a)+b=c4(b+a) 
=(c+0)+a. 

55. We now proceed to generalize the associativ 
law for addition in the following theorem. 

Theorem. // the order of the elements of a complex sum 
be kept fixt, its parentheses may be changed in any way. 

I. The sum of a standard sum and a single num- 
ber is a standard sum. Thus the sum of ((@+ 0) + 
c)+dand eis ((a+ 6)+c)+d)+¢e. 

II. We will now show that the sum of a single 
number and a standard sum is equal to the standard 
sum of the numbers involvd written in the same order. 

Let the given sum be 


s=a+{([O@+c)+ad]+---4+7)+h}. §6. 
Set Z=([(+c)+d]4+---4+7)+4&. 
Then s=a4+l, §§ 49, 34. 
Set for all of 7 except the last element 2, so that 
m=[(6+c)+d]+---+2 
Then l=m+k.- 
And s=a+t(m-+ bk). 








(1) The name ‘‘ Associativ Law’’ was given by Wm. Rowan Ham- 
ilton, Taylor’s Philosophical Magazine, 1844, p. 246. 


FUNDAMENTAL IDEAS. 21 


By the associativ law this givs 
S= (a+ m) + k. 


We thus hav £ written in the proper place for the 
standard sum. 

Now consider a+ m. Setting x for all of m except 
the last element 2, so that i 


u=([(6+c)+a]+--- 
and m=n +1, 
wehav a@+m=a+(u+7)=(a@+n)4+2 
and s=(@+m)+kh=((a4+n)4+7)+2. 

We may continue this process until the letter that 
stands in the inner parenthesis with @ represents only 
a single element. 

Thus 

s={([(@+) +e] +4) +--+} +8 


The proof is, then, complete and s is shown to. be 
equal to the standard sum of its elements, written in 
the same order. 

III. We will next show that the sum of two stand- 
ard sums is equal to the standard sum of all the num- 
bers involvd, written in the same order. 

Consider the sum 


Beet iy Vite sort 6) ta 
+ {([(g+%) +2] +---+2) +m}. 


If we set 


22 PRIMARY NUMBERS. 


n= (a+) +] tt) +h. 
so that 


sant (gt +i] 4-42) 4m, 


we can prove in the same way as befor that 
s={([m+e)t+A] 4+2)4+---4+4 4m. 


Substituting for 2 what it represents, we hav 


s={(((Le+ 9+ ]+-- +9 4+f} +e) 44] 
t+2)++---+2} 4 m. 

IV. Next consider any complex sum whatever. 
This must be built up from a number of simple sums 
by adding these sums to single elements or to each 
other. Each, of the simple sums we start with is a 
standard sum and therfor, by what has been proved 
above, the sum obtaind at any step of the process is 
equal to the standard sum of the same elements, writ- 
ten in the same order. Therfor the complete sum is 
equal to the standard sum of its elements, written in 
the same order. 

No matter, then, how the parentheses ar arranged, 
the given complex sum is equal to the standard sum 
of the same numbers, written in the same order. 
Therfor all the sums obtaind by varying the arrange- 
ment of the parentheses ar the same. 

56. Since a complex sum, the order of whose ele- 
ments a, 0, c, d, e,---is fixt, is independent of the 
arrangement of its parentheses, we may omit the pa- 
rentheses without ambiguity and speak of the sum 


FUNDAMENTAL IDEAS. 23 


até+tc+td+e4---, this expression being used 
to represent the sum obtaind by any arrangement of 
parentheses, 

67. Theorem. 7Zhe sum of the left members of any 
number of equalities 1s equal to the sum of the right 
members. 

hs VERE an VES Se eR 

The sum of the left members is a+c+e+g4.-.-- 

Substituting for a,c, e, g,---their respectiv equals 
We, fh. we Nav 


ge 


a@atetetgt+---=b4d4+ftht.-- § 52. 
58. We will now generalize the commutativ law 
for addition. 
Theorem. /2 any complex sum the order of the ele- 
ments may be changed in any way. 
Let the given sum be 
3 she ook TCT G tle te, «tla ey ie Ey A © aS Fa eat A 


and the form in which we wish to write it 


d+c+teta+d4... 
We hav 


ee eee a te te fees « 
=((@+---+64.---+c¢c4---)+¢@) 
Se gE i § 56. 
= (d+ (a4+---+6+---+0¢+4--)) 
Be ers eden id §§ 41, 46. 


24 PRIMARY NUMBERS. 


=dA4+((a+---+b4.---)40) 

Pee te b= i855. 
=At+(ct+(at---+O4+.---))f---fe4-:- 
= O(a ae 1 Ot). ee) tae 
=a2+e+(e late ee Pie) pee 
a (PrametRea gr a pared (a edly Se oo: 
=d+ctetat+(St+(--))t+--: 
=d+ctetat+b4... 


59. The theorems of §§ 55, 58 may be combined 
into one as follows : 

Theorem. Generalized Associativ and Commutativ 
Law for Addition. Jz any complex sum the arrange- 
ment of the parentheses and the order of the elements 
may be changed in any way. 

Or, a complex sum 1s independent of the arrangement 
of its parentheses and the order of us elements. 

60. Reverse of Addition. Separation into Parts. 

To get the sum of two given numbers a and 4 we 
joind two groups of @ and 6 objects to form a single 
group. Reversely, having given any group of objects, 
other than a group consisting of but a single object, 
we can divide it into two groups, which may be spoken 
of as parts of the original group, calld the whole. 
Similarly the numbers of objects in the two partial 
groups may be spoken of as parts of the number of 
objects in the whole group, this latter number being 
calld the whole. 


FUNDAMENTAL IDEAS. 25 


61. Axiom. Any number, except one, can be divided 
into two parts and is equal to the sum of those parts. 

62. Resolution into ones. Just as a number, not 
one, can be divided into parts, so each of its parts, not 
one, can be divided into parts and the process con- 
tinued until each part is the number one. We hav, 
then, the following theorem. 

63. Theorem. Axy number a, except one, 1s equal to 
the sum of a ones. 

64. Definition. On account of this property of the 
number one, that all other numbers are made up of 
ones, the number one is sometimes called unity. It is 
the unit or base upon which the system of numbers is 
built. For it the symbol 1 is used. 

65. Definition. This property of unity may be used 
to form a system of symbols and names for numbers. 
Thus we may let 2 stand for the sum of I and 1, 
3 for the sum of 2 and 1, and so on; thus 


see — Teel 
ae see Tat 
A= 3 ae Pt 


5=44+1 
6=5+1 
=641 
Ore 7 tnt 
9=8 + 1(’). § 38. 





(') For the origin of the symbols I, 2, 3, 4, 5, 6, 7, 8, 9 see Ball, 
p. 189. 


20 PRIMARY NUMBERS. 


We might continue this process as far as we chose, 
using a distinct symbol to represent the sum of each 
number and unity ; thus we might set 


Yes o5 a 
O=P+1 
S=O+1 
2=Ct+1 
P=2t+!1 
é=¥F4+1 


W=a@+1 


‘Lhe Mumbers 2-4 3s: 7, oy, (oe ee Oe 
¥ c, (0 an Named two, three sour alive, six seven, 
eight, nine, ten, eleven, twelv, thirteen, fourteen, 
fifteen, sixteen (°). 

Since any number @ is the sum of a ones, it is evi- 
dent that it would hav a place somewher in sucha 
series, if the series were carried out far enuf. 

66. The system of symbols and names above given 
has no rule connected with it by which the numbers 
represented by the symbols and names may be re- 
memberd. Thus, ther is no reason why the symbol 
7, or 9, should not be used instead of the symbol 5 to 
represent the number 4+ 1. The meanings of the 
symbols and names must, then, be memorized out- 
right. 


(1) See Pierce, ‘‘ Problems of Number and Measure,’’ 1898, p. 9. | 


FUNDAMENTAL IDEAS. 27 


It is evident that the further such a system is carried, 
the greater is the tax on the memory. Another sys- 
-.tem has therfor been devised, which avoids this diffi- 
culty. It will be explaind in a later article. 

67. Definition. The system of numbers I, 2, 3, 4, 

--, each of which is equal to the preceding plus one, 
is calld the natural series of numbers (’). | 

68. Definition. The alternate numbers, commenc- 
ing with one, in the natural series ar calld odd; the 
Glersseven (7). Wl hus: 1993; 5,--- ar odd ;- 2.4510, --. 
even. : 

69. Theorem. /f azs any number in the natural 
series, the numbers that tmmediately follow aara+t 1, 
a+2,@4+3,::-,@+n,a+(n+1),---, where 1,2, 
3,°°:, 2,n+71,-+- 0s the natural series of numbers. 

Let the numbers that follow @ in the natural series 
DeIDEOLA eo Mila. NEN 
b=ati § 67. 
ele (el oe eta Te) a fe 

| §§ 67, 46, 54, 65, 49. 
@—ce+i—(a+2)+i1=a+et+i) =a 4 3, 


Moreover, if 


k=at+yn, 
eee tt) Ei 2 (7a |, 
enceetnernumbers =), c,d... 4, 7.--- ar equal 


(1) See Stolz und Gmeiner, p. 13; Euler, ‘‘ Elements d’ Algébre,”’ 
edition of 1807, p. 7. 
(2) See Euler, p. 21. 


28 PRIMARY NUMBERS. 


respectivly to the numbers a+ 1,a+2,a+3,---, 
atn, at(n +1), > “Inothiseseres uthemn ampets 
added to a ar each, after the first, obtaind from the 
preceding by adding 1 to it. These numbers ar ther- 
for the natural series of numbers. 

70. Definition. Correspondence. If two groups of 
objects ar the same in number, we may in thot con- 
nect each object of one group with a definit object of 
the other group. Wear then said to make the objects 
of the two groups correspond to each other. 

If one of the groups contains z objects and the 
other is the group of z numbers of the natural series 
I, 2, 3, ---, 2, then, when the correspondence is made, 
the group of z objects is said to be arranged in order. 

As outward evidence of the connection between the 
objects and the numbers, the objects may be markt 
with the symbols I, 2, 3, -:-, z. 

The object corresponding to one is calld the first 
object, the object corresponding to two, the second, 
that corresponding to three, the third, and so on, that 
corresponding to z, the nth, or last object. 

Any portion of the natural series of numbers start- 
ing with I, as I, 2, 3, ---, 2, may be made to corre- 
spond to itself, each number corresponding to itself. 
Thus 1 is the first number in the series, 2 the second, 
and so on, z the last. 

71. Definition. Counting (") is the name given to 


(1\ The idea of number is prerequisit to, not derived from, the idea 
of cuunting. See Lefevre, p. 21. 


FUNDAMENTAL IDEAS. 29 


any process for finding the number of objects in a 
given group. 

This is usually done by separating the objects one 
after the other from the group and at the same time 
speaking, or thinking, the numbers I, 2, 3, 4, --- in 
succession until the last object has been removed. 
In this way we form a correspondence between the 
objects of the group and the natural series of numbers. 
The number spoken when the last object is removed 
is the number of objects in the group. This follows 
immediately from the definition of “sum” (§ 37) and 
the definitions of the numbers 1, 2, 3, 4, --- (§ 65). 

72. Theorem. Zhe number of numbers in the portion 
of the natural series of numbers starting with \ and 
ending with the number ais a. 

This is evident by counting. 

73. To add two given numbers @ and 4, we may 
write each number as a sum of ones and count the 
ones in both considerd as a single group. The re- 
sult is the sum of a and 4, 

Or we may find the 4’th number in the natural series 
after a2. This number is a + 6 (§ 60). 

74, Addition Table('). In order to make a table of 
the sums obtaind by adding all possible pairs of the 
numbers I, 2, 3, ---, (0, except those sums which may 
not be represented by the symbols so far defined, we 
draw a square with its sides parallel to the edges of 
the paper. Then we divide the top and left-hand sides 





(1) See Tannery, p. 40. 


30 PRIMARY NUMBERS. 


each into (0 equal parts and thru the points of division 
draw lines parallel to the sides forming rows and 
columns of little square compartments inside the given 


square. 
The divisions of the top and left-hand sides we mark 
on the outside with the numbers I, 2, 3,---, (0, start- 


ing at the upper left-hand corner. These numbers 
giv the order of the rows and columns. 

Now we wish to place in each compartment the 
number which is the sum of the two numbers which 
ar found respectivly at the left of the row and at the 
top of the column in which the compartment lies. 

Thus in the first row inside the given square we 
wish to write the numbers that represent the sums 
I+1, 1+2, 1+3,--- But these ar the same as 
the numbers of the natural series following 1 (§ 69). 
Similarly in the second row we write the numbers in 
the natural series after 2, and so on. 

It will be noticed that this table, which is given be- 
low, is symmetrical with reference to the diagonal line 
starting at the upper left-hand corner and going to the 
lower right-hand corner. This is due to the commu- 
tativ law for addition. 

This diagonal line is calld the principal diagonal. 


vi 


31 


FUNDAMENTAL IDEAS. 


ADDITION TABLE. 




































































713191 7/016) 2) #161 
819 /P/ 0/6) 2) F134) 0 











2/3/4/516l7|8lolPlolSi2|¥ie\n 








314/5|6|7/8|9/P/0/E| 2) ¥] 4) 10 











8/9 /P/0|B|2|¥| 2] 10 
91/0] 6|2|¥] ea] 








vie 
8 


we 
7 


8lolrlol|GBl2|¥i 4/0 





9|F/0) 8] 2|¥) 3] 0 
P10} G\2/¥/ 3/10 
0|6|2|¥| 3/0 
S121 ¥)/ ea] 0 
2|¥|e| 0 

















G 














CHAP LE Rigs 


SUBTRACTION. 
GREATER. LESS. DIFFERENCE. 


75. Definition. Having given two numbers a and 
6, if a + 0, ther is some third number + such that 
eithera=xr+d0orr+a=d. S45; 

In the first case a is said to be greater than J, in the 
second a is said to be less than 0(’). 

More explicitly, in the first case a is said to be x 
greater than 4 and in the second case x less than 0. 

Thus, since 8 = 3+ 5, 8 is greater than 5; since 
2/-Fab = 5, 5 is less'than 3: 

The statement ‘a is greater than 0” is written 
a>; the statement ‘a is less than 0” is written 
a<90(’). 

The definitions of the terms ‘greater than’’ and 
“less than’’ may, then, be stated in symbols: 

If a= «+, then a> 46; and conversely, if a> 43, 
then a=27-+ J. 

If ++ a= 4, then a <6; and conversely, if a <= @, 
than r+ a= 0. 





(1) See Tannery, p. 8; Hadamard, ‘‘ Lecons de Géométrie €lémen- 
taire,’’ p. 46. , 

(2) The signs >>, < seem to hav been used first by Vieta. See his 
‘In Artem Analyticam Isagoge,’’ 1591. 


32 


SUBTRACTION. 33 


The statements a>6 and a<8@ are called in- 


equalities. 
76. Theorem. Jf a=x+6 and ++ y, then 
apy td. 
For, sincer+y,xr+b04+y+ 0. § 47. 
Hence, sincea=r+b,at y+. Sous 


77. Definition. From the last theorem we see that 
if~=2x+ 4, ther is no number y, different from 1, 
such that a= y + 4. 

Hence, if @ and dar given, x is determind by the 
equality a= 2+ J. 

This passiv number +, which depends only on a@ 
and 6 and is such that a=xr+ 4, we will call the 
right-handed difference between a and 0. 

This difference may be represented by some symbol 
showing the two numbers from which it is formd. 
The symbol a — 4, read ‘“‘@ minus 4,” is customary (’). 

The difference a— 0 is, then, univalent, when it 
exists, that is, when ther is some third number +, 
such thata=x+ 0. 

The number a may be calld the first element of the 
difference and 4 the second element. 

78. Theorem. Jf the right-handed difference a—O6 
exists and ts the number x, thena—b=x. 

This follows from § 6. 

79. Theorem. /fa=+x+40, then a—b=x, and 
conversely, ffa—b=x, thna=x-+ 0(’). 

(1) The sign — appears first in Widman’s Arithmetic. See foot-note, 
p- 13. It was first used in the modern sense by Stifel. 

(2) See Tannery, p. 15. 

3 


34 PRIMARY NUMBERS. 


80. Theorem. /f ++ a=6 and x+y, then 
ytat b, 

81. Definition. From this theorem we see that the 
equality x + a= 0determins the number 4 We will 
call this number the right-handed remainder of 0 less 
a, which we will represent by the symbol 4| a, which 
May be reads. Gless a. 

The remainder 4| @ is, then, univalent, when it ex- 
ists, that is, when ther is some third number x, such 
that r++a= 4. 

82. Theorem. Jf the right-handed remainder b|a 
exists and ts the number x, then x = b| a. 

83. Theorem. //4+ a=), thenx=6|a,; and con- 
_ versely, of x= b|a, thnx+a=d, 

84. Definition. Implicit and Explicit Definitions. 
Ther ar two classes of definitions. A definition of 
the first class merely givs a name to an idea with 
which we ar already familiar, usually a fundamental 
idea. Such definitions ar calld implicit definitions. 
A definition of the second class expresses an idea in 
terms of ideas simpler than itself, either fundamental 
ideas or ideas that hav already been defined in terms 
of fundamental ideas. Such definitions ar calld ex- 
plicit definitions. 

In §§ 75, 77, 81 we nav examples of explicit defi- 
nitions. The ideas ““ greater) than) ye -Siess= thar: 
“right-handed difference,” and ‘right-handed re- 
mainder” ar exprest in terms of ‘“ equality” and 
“ addition.” 


SUBTRACTION. 35 


An idea thus exprest is calld a derived idea. 

85. We will now prove some theorems connecting 
the ideas represented by the symbols >, <, —, |, 
which hav been independently defined. 

86. Theorem. /f a> 4, the right-handed difference 
a — 6 exists, and conversely. 


For, ifa@>),a=2x+4+ 4, § 75, 
Therfor the difference a — 6 exists and is the num- 
ber x. WME 


The converse theorem is proved by reversing the 
steps. 

87. Hence the symbol @ — das yet has no meaning, 
unless a> 8, 

88. Theorem. /fa< J, the right-handed remainder 
b | a exists, and conversely. 

This is proved in the same way as § 86. 

89. Theorem. /f a> b,. then 6 < a, and 
a|b=a—0; and conversely, if 6 <a, then a> b(') 
anda—b=a| db. 

Mion i @ > 0.a— 2-4 6and@—&= 2%, §§ 75, 77. 


Hence ++ 6=aand r= ald. §§ 7, 81. 

Therfor 6< a and a|6=a-— 4, §§ 75, 34. 

The converse theorem is proved by reversing the 
steps. 


It should be noticed that the proof of this theorem 
depends on the commutativ law for equality. 

90. In virtue of the last theorem, whenever we 
hav either of the statements a> 0, <a, we may 


(1) See Stolz und Gmeiner, p. 6. 


36 PRIMARY NUMBERS. 


replace it by the other and may replace either of the 
expressions a — 0, a| 6 by the other. 

We will find it convenient in future never to use the 
symbol |, but to use in its stead —. 

The symbols > and < will, however, both be use- 
ful, tho we will employ chiefly >. We should always 
bear in mind that the statement a > 6 may at pleasure 
be replaced by the statement 0 < a. 

91. Definition. Having given two numbers a and 


b, if a+ 4, ther is some third number 4 such that: 


eithhera=6+ 7rora+xr=8. §§ 45, 41. 

In the first case a may be said to be larger than 4, 
in the second case smaller than 0. 

The statement “a is larger than 6”’ may be written 
a> 6; the statement ‘ais smaller than 6” a< 3. 

The definitions of the terms “larger than” and 
“smaller than”’ may, then, be stated in symbols: 

Ifa=6+44, then a> 0; and conversely, if a> 4, 
then a=04+4 x. 

Ifa+x£= 4, then a <0; and conversely, if a< 6, 
thena+r=4, 

92. Theorem. /f/a=b+ rand x+y, thena+b- y. 

93. Definition. From this theorem we see that the 
equality a= 6+ x determins the number +. 

This activ number 4, which depends only on 6 and 
a and is such that a=6+4 x, we will call the left- 
handed difference(*) between 4 and a. For this differ- 





(1) See Grassmann, ‘‘ Ausdehnungslehre,’”’ edition of 1878, p. 5; 
Stolz und Gmeiner, p. 5 ; Schubert, ‘‘ Encyklopidie,”’ p. 9, 


SUBTRACTION. a7 


ence we will use the symbol 4 ~ a, which may be read 
“O from a.” 

The difference 0 —a@ is, then, univalent, when it 
exists, that is, when ther is a third number +x, such 
thata=04 4%. 

94, Theorem. Jf the left-handed difference 6 ~ a ex- 
asts and 1s the number x, thenb-a=x. 

95. Theorem. /f a=()+4 4, then b6-a=x, and 
conversely, fp b-a=x,thna=b+ x. 

96. Theorem. /f/a+r2=bandr+y,thena+y+0. 

97. Definition. From this theorem we see that the 
equality a@+2=06 determins the number x We 
will call this number the left-handed remainder when 
ais taken from 4. For this remainder we will use 
the symbol a |. 4, which may be read “a sul 0.” 

The remainder a |-0 is, then, univalent when it 
exists, that is, when ther is a third number +, such 
thata t+ r=. 

98. Theorem. Jf the left-handed remainder a | 6 
exists and ts the number x, then x =a |- 6. 

99. Theorem, /fa+24=0, then x=a|- 6b; and 
conversely, ff r= a} 6, thna+t+«r=d, 

100. Theorem. Jf a > J, the left-handed difference 
b — a exists, and conversely. 

101. Theorem. // a < J, the left-handed remainder 
a |. 6 exists, and conversely. 

102. Theorem. /f a> 6, then a> 6b ana 
6—-a=a—b, and conversely, if a> b, then a>b 
and a—b=b—a. 


38 PRIMARY NUMBERS. 


The proof of this theorem depends on the commu- 
tativ law for addition. 

103. Theorem. J/f a < 0b, then a < 6 and 
a|6=b—a, and conversely, if a< b, thn a< 6b 
and b—a=a|- 6. 

The proof of this theorem also depends on the 
commutativ law for addition. 

104. Definition. In virtue of the last two theorems 
we will in future never use the symbols > and <, but 
will use in their stead > and < respectivly. Simi- 
larly instead of 6 ~— a and 4 |: a we will use a — 0. 

In virtue of §§ 89, 102, 103 we will in future instead 
of the four terms “ right-handed difference,” ‘“ right- 
handed remainder,” ‘left-handed difference,’ and 
“left-handed remainder’’ use the single term “ dif- 
ference.” 

105. Theorem. No number is greater than itself. 

For, since a= a,a+24-+4a. § 43. 

106. Definition. The symbols > and < ar used 
to mean “is not greater than”’ and ‘is not less than” 
respectivly. 

The expression @=0 means that either a= 0 or 
a>b6, Similarly the expressions a=é, a=, etc., 
ar defined. 

107. Definition. Complex Expressions containing 
Plus and Minus Signs. The difference of two num- 
bers may be connected with other numbers by plus or 
minus signs, provided always that the first element of 
every difference is greater than the second element. 


SUBTRACTION, 39 


Thus we may hav expressions like [(a—6) +c] —(d—e) 
or {[(@ + 8) —c] +d} —(e+f). 


All such expressions ar univalent, since every sum 


and every difference is univalent. S81007 7 
108. Theorem. 2+ 6>6(')and (a+ 6)—d=a(’). 
For PE Ba 


Therfor a+6>6and(a+6)—b=a. 8§75, 79. 


109. Theorem. A whole is greater than each of tts 
parts. 

110. Theorem. Jf a> J, then (a—6)+0=<a. 

Since a>b,a=x+banda—b=-2. §§75, 70. 


Hence (a—6)+b=2+6. 9§ 46. 
Therfor (a—6)+b=<a. § 34. 


111. Theorem. Jf a>3d, then a+c>b+e and 
(a+c)—(6+c)=a—b , and conversely, ffatc>bte, 
thna>banda—b=(atc)—(O+ 06). 

For the direct theorem, 


since a@>6,a=x*+6 and a—b=-%. 8§ 75, 79. 
Hence atce=(x++6)+c=4+(b4+c) §§$ 46, 54. 


and (a+c)—(6+c)=-. 8$§ 34, 79. 
Therfor ate>b+e § 75. 
and (a+c)—(6+c)=a—d. § 34. 


(1) See Stolz und Gmeiner, p, 15. 
(2) See Schubert, ‘‘ Encyklopidie,’’ p. 9 ; Tannery, p. 44. 


40 PRIMARY NUMBERS. 


For the converse theorem, 
since @atc>b64+oate=rx4+ (64+c)=(x+ b) +¢ 
and (a+c)—(6+4+c)=-%. 
Hence a=x+6b and a—Jd=x. 
Therfor a>banda—b=(a+c)—(6+4+ 0). 
112. Theorem. /f a> 0, then c+a>c+6 and 
(¢ + @a)—(¢c+ 6)=a—6; and conversely, of 
c+a>c+), thena>b and a—b=(c+a)—(c+4). 
113. Theorem. //a=db and b> c, thena>c and 
a—c=b—c, and conversely, if a>c, b> c, and 
a—c=b—c,thna=b. 
For the direct theorem, 
Since 2. | 6b=x+c¢ and 6—c=¥~7. 
Sp Any ibe 


Hence, since a=0, a=az+c and a—c=7. 
Therfor a> 1c and a — C— 0 — 


For the converse theorem, let a—c= yz. 
Then 6—c=42. 


Hence a=x+ec 
and b=x+e. 
Therfor a= b. 


114. Theorem. /fa>dandb=c, thena>c and 
a—b=a—cy, and conversely, if a>b, a>c, and 
a—b=a—c.thnb=c. 


Lal 


SUBTRACTION. 4I 


For the direct theorem, 
since a> 8, a=xt+6O and a—bd=x 
Since ere ttb=xrt+e. 
Hence @=x+c and a@—c=x%. 
Therfor Ge eal = 


For the converse theorem, let a—d= 4, 
Then @—c= 42. 


Hence a=xt+6 
and | a=x+e. 
Hence KHtb=44+. 
Therfor b=c. § 49. 


115. Theorem. /f/a>dandb>c, thena>c and 
a—c=(a—6)+(b—c). 


eee a> 0, 1: a=xt+06 and a—bd=-x. 
since 4 > c, b=y+te and b—c=y. 
Hence t+barii(yte). § 4o. 
Hence a=(*+y)+e and a—c=xr++y, 


Therfor a@>c and a—c=(a—6)+(b—c). $50. 


‘116. Theorem. Jf we hav a series of inequalities of 
the same kind,asa>b>c>d>e>.--., any number 
in the series 1s greater than any following number. 

The proof of this theorem is identical in form with 
that of § 36. 


42 PRIMARY NUMBERS. 


117. Theorem. Jf we hav a series of equalities or 
inequalities of the form aSbScSadsSe5..., any 
number in the series 1s equal to or greater than any fol- 
lowing number, the sign being >, of ther ts at least one 
sign of inequality in the series between the two numbers. 

The proof is like that of § 116. 

118. Theorem. /f a@ > 6 and b> c, then 
a—c>b—c and (a—c)—(b—c)=a—b, and 
conversely, if a>c, b>c, and a—c>b—c, then 
a>b anda—b=(a—c)—(b—Cc). 

For the direct theorem, 


since a>b>c,a>canda—c=(a—d) + (b— Cc). 


§ 115. 
Therfor (fed pee EN 
and (a —c) —(@—c) = a0. §§ (75; 70. 

For the Beriverce, theorem, 

Since Wi, @a=*+e and a—c=xX. 
Since Ga "c. b=y+e and b6—c=y¥,~ 
Since a—c>b—c «>y. ‘. § 117. 
Pilencé ate>yte. Siig 
Or Mae vee See or 


Therfor, by the first part of the theorem, since 
a>b>c,a—b=(a—c)—(b—- Cc). 

119." Theorem. J/f a > 6b and FOR ier hex ‘ 
a—c>a—b and (a—c)—(a—b)=b—c; and 


SUBTRACTION. 43 


conversely, if a>¢, A>; and a—c>a—b, then 
b>c and b—c=(a—c)—(a—d). 

120. Theorem. Jf a > 6 and c = d, then 
atc>b+d and (at+c)—(64+d)=a—4; and 
conversely, ff at+c>b+d and c=d, then a>b 
and a—b=(a+c)—(64 24). 


For the direct theorem, 
sinceea>batc>bte 
zhao (a+c)—(6+4+c)=a—A6,-§ 111. 
Sincec=d,b6+c=64d. § 409. 
Therfor OQ oo Ua. 
and (a+c)—(64c)=(@+c)—(64+ 2a). § 114. 
Also (a+c)—(6+d)=a—Q4, § 34. 
For the converse theorem, 
sincec=d, bO+c=64d. 
Hence, since a+c> 0+ d, 


Ws (oe (ge 8 
and (a+c)—(64+d)=(a+c)—(6+0)9 
Therfor a>6b and a—b=(a+c)—(6+0). 
Also a—b=(a+c)—(6+2). 


121. Theorem. /f a>J6b and c>d, then atc 
>b+dand(a+c)—(64+d)=(a—4)4+(c—-2@). 
Sinced. >> 0, 


A4 PRIMARY NUMBERS. 


a@+te>b+cand(a+c)—(64c)=a—0., §111. 
SInceic. > a, 
6+c>6+4+dand (64+ c¢c)—(64+d)=c—d. § 112. 
Therfor a+c>6+d and (a@+c)—(64+d)= 
(e@+)—G+o] +[6+9-C+4)] 
=(a—6)+ (c—2@). §$ 115, 50. 
123, theorem. if a= 0; c—d, andb >\c tex 
a>c, b>d, and a—c=b—d; and conversely, of 
a>c, 6>d, a—c=b—d, and c=d, then a=b; 
tia>c,o6>d,a—c=b—d, and a=), then c=d. 
For the direct theorem, 
sincea=b>c, a>c and a—c=Jb—-e. 
Since 6>c=d, b6>d and 6—c=b—d. 
Therfor a—c=b—d. 
For the first converse theorem, 
sinceb>d=c, b6—d=b—c. 
Hence, since a—c=b—d, 
a—c=b—c. 
Therfor a= db. 


Similarly the remaining theorem may be proved. 

123. Theorem. /fa>d,c=d,and b> c, thena>c, 
b>d,a—c>b—d,and(a—c)—(6—d)=a—d; 
and conversely, ff a>c, b>d, a—c>b—d, and 
c=d, thena> 6b. 


+ 


SUBTRACTION. 45 


For the direct theorem, 


since a>bd> 6, DI i 
a—c>b—c, and (a~—c)—(6—c)=a—ZB, § 118. 
Since 6>c=d, b>a 


and b—c=b—d. 
herior a— co > 6— a 
and (a—c)—(6—c) = (a—c) — (6-2). 
Also (a—c)—(6—d)=a—ZJ, 
For the converse theorem, 


since 6>d=c, Oe) et a 


Hence, since a—c>b—d, 
a—c>b—c. 
Therfor a>b § 118. 


124. Theorem. J/f/a=0,c>d,and b> c,thena>c, 
ewe — a) —d,and (bd) —(@—c) =c—a: 
and conversely, ff a>c, b>d, a—c<b—d, and 
a=b, then c>d. 

For the direct theorem, 


since a=Jd> 6c, (9 Meeps 
and a—c=b—<c, 
Since 6>c> d, b> d, 
b—d>b—c, and (6—d)—(6—c) =c—d. 


46 PRIMARY NUMBERS. 


Therfor a—c<b—d 


and (6—d) —(a—c) =(6—d)—(6—0). 
Also (6—d)—(a—c)=c—d. 
For the converse theorem, 
since J= 24> ¢, peice 
Hence, since a—c<b—d, 
b—c<b—d. §§ 89, 114. 
Therfor CO! Sai hey 


125. Theorem. /fa>b,c>d,andb>c, thena>d, 
a—d>b—c,and(a—d)—(b —c)=(a— 6)+(c —2). 


Since\ Ge c,d. oa: 
b—d>b—c, and (6—d)—(b—c)=c-d. 
Since Z > o-> ad, tea, 


a—d>b—d, and (a—d)—(b—d)=a-—Qb, 
Therfora—d>6b.—c and (a—d)—(6—c)= 
[eid OO) |e ee as 
=(a—b)+(ce—d). §$§ 115, 50. 
126. Theorem. /fa>d,c>d,anda—b=c—d, 
then a+d=b+c; and conversely, ff a>b and 
atd=b+c¢, thenc>danda—b=c—d. 
For the direct theorem, 
since a> 4, Go a > Uae 


and (a+ da)—(64+d)=a—d. Sets 


SUBTRACTION. 47 


Since c>adab+ec>b4+d 


and (64+ c¢c)—(064+ d)=c—d. § 112. 
Hence, since a—b=c—d, 

(a+ d)—(6+d)=(6+c¢)—(64+ 2). 
Therfor - atd=b+e. aS Vtee 


For the converse theorem, 


since a> 0, ad > ba 


and (a+d)—(64+d)=a—Q4, 
Hence, since atd=db46, 

6+c>b+a 
and (6+c)—(64+d)=(a+d)—(64 4). § 113. 
Fience Ca 
and c—d=(6+c)—(64+24). Salas 
Therfor a—b=c—d. 


127. Definition. From the preceding theorem it fol- 

lows that if either of the statements “a> 6, c > d, and 
peo and “‘as> 0 and a+ d—6-+-c’) 1s 
true, the other is. The truth of either statement is 
then said to be a necessary and sufficient condition 
for the truth of the other. The truth of either state- 
ment follows necessarily from the truth of the other ; 
and the truth of either is sufficient to establish the 
truth of the other. 


48 PRIMARY NUMBERS. 


Whenever a theorem and its converse ar both true, 
we have two statements, the hypothesis and conclusion 
of either theorem, either of which statements is a 
necessary and sufficient condition for the truth of the 
other. 

128 Theorem. /fa>d,c>d,anda—b>c—d, 
thena+ad>b+cand(a+d)—(6+c)=(a—b)— 
(c—d); and conversely, ff c>d anda+d>b+e, 
then a>banda—b>c—d. 

For the direct theorem, 


since a > 8, AP Bed ays 
and (a+d)--(64d)=a—. 
Since c > d, BSD ek ME yh 
and (64+ c)—(64d)=c—-d. 
Hence, since a—b>c—d, 


(a+ d)—(64+4)>(6+—(644) 
and [(a + 2)— (6+ 4)]—[(6 + ¢)— 6 +4) 
=(a—6)—(c—@). 
Therfor atidSPae 
and (a+ d@)—(6+c)=(a—b)—(c—2@). 
For the converse theorem, 
since ¢ > d, bac Sip ae 
and (6+¢c)—(64+da)=c—d. “ 


SUBTRACTION. 49 


Hence, since atd>b-+e, 

: Gr ea ogy! 
© and (a+d)—(64+d@)>(64+c)—(64 2). 
Therfor a> danda—b=(a+d)—(b64+ 2). 
Therfor a—b>c—d. 

129. Theorem. Having given any two numbers a 

and b, one of the statements a=b, a>b, and a<b 
must be true, and only one can be true (’). 


For either a = 4 or a + 8, 
In the latter case either a=xr+d0orr+a=d. 


§ 45. 
Hence either a> 6 ora < 8, 
Therfor one of the statements a= 6, a> 6, and 
a< 6 must be true. 
Now, ifa=da+tr+dandr+az d, 


8$ 43, 7, 8. 
Hence, ifa=b,apbandact J. 
lia = 0, a=x+ 2, 
Hence as dO. § 44. 


Moreover, if a>d,a<6. For if a< 34, since also 
a> 6, we would hav a> a ($115), which contradicts 
$105. 

Hence, tf a> 6,a+ anda + 0. 

Similarly it may be shown that if a< 6, a+ 6 and 
apd. 

(1) See Stolz und Gmeiner, p. 6. 

iy: 


50 PRIMARY NUMBERS. 


Therfor only one of the statements a = 0, a> 6, and 
a <%$can be true. 

This theorem may be more briefly, tho less exactly, 
stated : 

Having given any two numbers a and b, either a= 6, 
Ca OOF OO: 

130. Definition, Every proof, when written in its 
complete form, consists of one or more steps calld 
syllogisms('). Every syllogism consists of three state- 
ments, the major premis, minor premis, and conclu- 
sion(*). The major premis is always a general truth 
which has already been establisht, either a definition, 
axiom, or theorem. The minor premis is a particular 
truth from which and the major premis the conclusion 
follows as a necessary consequence. 

131. Ther ar several kinds of syllogisms, but the 
ones most frequently used in mathematics ar five in 
number, two in which the major premis is a categorical 
statement, two in which the major premis is a hypo- 
thetical statement, and one in which it is a disjunctiv 
statement. These may be represented symbolically as 
follows: 


I. CoNSTRUCTIV CATEGORICAL SYLLOGISM. 
Major Premis. All Bis C. 
Minor Premis. Ais B. 
Conclusion. Therfor A is C. 


(1) See Jevons-Hill, p. to. 
(2) See Jevons-Hill, p. 113. 





ve 


SUBTRACTION. BI 


II. DEstructTiv CATEGORICAL SYLLOGISM. 
Major Premis. All Bis C. 
Minor Premis. A is not C. 
Conclusion. Therfor A is not B. 


III. Constructiv HypoTHETICcAL SYLLOGISM ('). 
Major Premis. If A is B, then C is D. 
Minor Premis. Ais B. 
Conclusion. Yherfor Cis D. 


IV. Destructrv HypoTHETICAL SYLLOGISM. 
Major Premis. If Ais B, then Cis D. 
Minor Premis. Cis not D. 
Conclusion. Therfor A is not B. 


V. Disyunctiv SyLiocismM(’). 
Major Premis. Either A is B, or C is D. 
Minor Premis. A is not B, 
Conclusion. Therfor Cis D. 


132. Definition, When a constructiv syllogism is 
used in a proof, the proof is said to be direct; when a 
destructiv syllogism is used, the proof is calld indi- 
rect, or a reduction to absurdity (*); when syllogism V 
is used, the method of proof is calld the method of 
exhaustion or exclusion. 

Up to § 129 all our proofs were direct. But in that 
article we used an indirect method to show that if 
go>, then: a) <td. 

(1) See Jevons-Hill, p. 151. (2) See Jevons-Hill, p. 157. 

(3) See Beman and Smith, ‘‘ New Plane and Solid Geometry,”’ 
IQOI, p. 14. 


52 PRIMARY NUMBERS. 


133. Theorem. /7 the natural series of numbers each 
number ts one less than the immediately succeeding num- 
ber. 

For, if a+ 1 = 4, ais one less than J. Si7 53 

134. Theorem. J/x the natural series of numbers each 
number ts less than every succeeding number (*). 

135. Theorem. Uvity zs the smallest of all numbers (’). 

For unity is the first number in the natural series 
and every number has a place in that series. 

136. Theorem. Having given any number, ther ts 
a number greater than tt. 

For, if @ is any given number, the number @ + I is 
greater than it. 

This theorem is equivalent to the statement that 
ther ts no greatest number, 

137. Definition. Infinit. Finite. When, having 
given any number of objects of a particular class, 
another of the same class can be found, the class is 
said to hav an unlimited, or infinit number, of objects ; 
less exactly, the number of objects in that class is said 
to be unlimited or infinit. 

In contrast, the number of objects in a group, which 
does not hav the above property, is said to be limited, 
or finite. 

188. Theorem. Zhe number of numbers is infinit (?). 





(1) See Cantor, ‘‘ Sur les fondements de la Théorie des Ensembles 
Transfinis,’’ translated from the German by Marotte, 1899, p. 14. 

(2) See Cantor, p. 17. 

(*) See Lucas, ‘‘ Théorie des Nombres,’’ 1891, p. 3. 


+f 


SUBTRACTION. 53 


For, having given the numbers 1, 2, 3, ---, a, ther 
is a number a + 1 greater than all of these (§ 134) 
and therfor different from them. ae -zeP 


The natural series of numbers has, then, no limit. 
It has a beginning, but no end. 

139. Theorem. /f a<d<c, 0 ts one of the num- 
bers in the natural series between a and c. | 


Let d=a4ux. 

The numbers following a in the natural series ar 
a+iI,a+2,a+ 3,--- § 60. 

Since + is a number in the series I, 2, 3, --- ($65), 


one number of the above series is a + # or 0. 

Hebi == 30-4, 

Then 6+ yor c is one of the numbers 6+ 1,64 2, 
6 + 3, ---, which follow 4 in the natural series. 

We hav then the portion of the natural series a, 
@at+1,@+2,a+3,--,6641,642,064 3,---,¢. 

Therfor 4 is one of the numbers in the natural series 
between a and c. 

140. Definition. On this account, when a<d<ce, 
6 is said to lie between a and ¢ in magnitude. 

141. Theorem. Jf a and b ar any two consecutiv 
numbers in the natural series, ther exists no number be- 
tween a and b (’). 


For, suppose a hentid /p 
Then b—a>b—cE1. §§ 119, 135. 
Hence b—a>l. 


(1) See Cantor, p. 15. 


54 PRIMARY NUMBERS. 


But by hypothesis 6=a@-+ I. 
Hence b—a=l. 


Since 6—a cannot be both greater than one and 
equal to one (§ 129), the theorem is establisht. 

142. Subtraction. When two unequal numbers a 
and & ar given, we may determin which is the greater 
and find their difference, a — 6 or d—a, by the fol- 
lowing method. 

Consider the two series of numbers 


6b, b+1, 642, 643, ---, b+a, 

a, a+iI, a+2, a+3, -:-, a+, 
Now either Zp OU eeu. § 1209. 
In the first case b<a<b4a. § 108. 


In the second COR AAT TD: 


In the first case, then, a must be one of the num- 
bers in the first series between 6 and 64 a. § 139. 

In the second case 6 must be one of the numbers in 
the second series between a and a + &@. 

In any case, then, either a will be some number 
6+ xin the first series and the difference a — 6 will 
be 4, or 6 will be some number a + yin the second 
series and the difference 6 — a will be y». 

By forming these series, therfor, we may determin 
whether a> 6 or a< 0 and find the difference a — 4, 
ifa> 0%, or b—a, ifa<J. 


SUBTRACTION. 55 


143. Thus, to find whether 5 > 2 or 2 > 5, and also 
to find the difference between these numbers, we form 
the two series 


Shoe ar te ee 
and 2, 2+1, 2+2, 2+3, 2+4, 245, 
Or 5) 6, if 


and 2, 3) 4, 5, 6, 7: 


We find that 2 is not in the first series, but 5 is 
inetne seconds and 5 = 2-4-3, Hence 5.> 2 and 
5—2= 3. 

Hence also the symbol 2—5 is meaningless. 
Siow 

144, Theorem. <Axy given set of numbers, no two of 
which ar equal, may be arranged in a series so that each 
shall be less than the succeeding one. 

Merticraivenssctebe a 407 Cad te, ac 

To arrange these numbers in a series so that each 
shall be less than the succeeding one, we put down 
any one number first, say a. 

Then we compare any other, as 4, with a. 

If 6 <a, we set down @ in front of a. If 6> a, we 
set 4 down after a. 

ip0ses ( s.78 wLhen we hav sthe series 2,70, 
wher a < @. 

Next we compare c with a and place it befor or after 
a according as it is less or greater than a. 


56 PRIMARY NUMBERS. 


Suppose c> a. Then we must compare c with 0. 
lic =< & we hav the series a, c.0, wher a = 6-0: 

_ Similarly we may compare each of the other num- 
bers in turn with those already in the series and assign 
to it a place in the series. 

Thus we may hav the series a, c, 0, d, e,---, wher 
Oe) ee 

Numbers so arranged ar said to be arranged in 
order of magnitude, or simply orderd. 

145. Definition. Operation of Subtraction. The proc- 
ess described in § 142 for finding whether a> 0 or 
a <6 and for finding the difference between a and J, 
or between @ and a is calld subtraction. 

Thus the difference a — J, when it exists, is obtaind 
from a and & by an operation, the operation of sub- 
traction. In this operation @ is the operand, the 
passiv element, or minuend, and 4 the operator, the 
activ element, subtrahend (*), or subtrahent (’). 

We may also obtain a — 6 by the use of objects (’). 
If from a group of a@ objects we take away @ objects, 
the number remaining will be a — 8. 

Or, if from a group of a objects we take away enuf 
objects for the number remaining to be 4, the number 
taken away will be a — 4, 

These results follow directly from § IIo. 





(1) See Schubert, ‘‘ Encyklopadie,’’ p. 9. 

(2) This name, with an activ ending, was proposed by Schrdéder. 
See Schubert, ‘‘ Mathematical Essays,’’ p. II. 

(3) See Tannery, p. 13. 


+? 


SUBTRACTION. 57 


146, Subtraction Table. In order to make a table 
of the existing differences between all possible pairs of 
the numbers 1, 2, 3,---, (0, we divide a square into 
compartments as in § 74 and write the numbers 
-I, 2, 3, ---, 0 along the topand left-hand sides of the 
square as in that article. 

In each compartment we wish to place the number 
which is the difference between the two numbers found 
respectivly at the left of the row and at the top of the 
column in which the compartment lies, if this differ- 
ence exists. 

Thus in the sixth row of compartments we should 
write the numbers that represent the differences 6 — 1, 
Ol 2) 0. — 351: - 

To find these differences we may use the addition 
table (§ 74) to advantage. We notice that in the com- 
partments of that table the 6’s lie in a line going from 
the top downwards and to the left. The 6 of the first 
row is in the fifth column; hence 6—1= 5 (§ 709). 
The 6 of the second row is in the fourth column; 
hence 6—2=4. Andsoon. Ther is no 6 in any 
row below the fifth, so that the differences 6 — 6, 6 —7, 
6 — 8,--- do not exist. 

Hence the sixth row of our new table will contain 
the numbers 5, 4, 3, 2, I in the first five compart- 
ments and none in the others. 

In a similar manner we may form the rest of the 
table, which is given below. 


ee 


58 PRIMARY NUMBERS. 


SUBTRACTION TABLE. 

















































































































AEE 2 SSS OWA Sd On ENO IE CH CaO! 
I 

PR aX 2 timieiad 

£33] 27 ete] eid We Pe hier i 
AE ea rie 
safa/2 [1 sl 
Ol t59 4a gee 

7) Ol Sites omer PAS ae fh nal ae 
Shu ON a a Ais eo may 

O18 POs Na ise ane ee i 
P19) 817) Oe) 4 i ayaa SS eee 
O18) 913) 71.615) 4735 cra Sa ee 
C/O) P0183) 7/6 (yal a2) ie ieee 
21 CLO | 918 376 1a 3 oli Tee ee 
F12/G/O(F)9/ 8/7) 6) 5/4] 3} 2}z, | [7 
G(¥/2(Cjol rio] 87/6] 514] 3)2irl | 
O(S|F/2;Bjo;rio/8/ 7/6/54) 3/2 














147. Definition. An inverse of a given operation is an 
operation that undoes the work of the given operation. 
148. Theorem. Zhe operation of subtraction is inverse 


to that of addition (’). 


For, as we hav seen in § 108, if we add 2 to a given 
number and then subtract 4, we get back the original 


number. 


149. Theorem. Addition ts inverse to subtraction. 
For (§ 110), if we subtract 4 from a given number 


and then add 4, we get back the original number. 





(1) See Schubert, ‘‘ Encyklopadie,’’ p. 8. 


ae 


SUBTRACTION. 59 


150. Definition. The operation of addition is calld 
a direct ('), or synthetic operation. Having given 
two numbers we find their sum directly, that is, with- 
out the intervention of any other operation. 

On the other hand, in subtraction, having given two 
numbers, of which the second is a part of the first, the 
object is to find the other part, that is, to find a third 
number which added to the second makes the first. 
Every question in subtraction is, then, turnd into a 
question in addition. 

For this reason subtraction is calld an indirect or 
analytic operation. 

For the same reasons as above given a+ 6 and 
a—oar calld respectivly synthetic and analytic com- 
binations (°) of the numbers a and 4. | 

The opposition in nature between the two opera- 
tions may also be well shown by the equality 
c= a+, 

If @ and dar given to find c, we get the result by 
addition. | 

If ¢ and 4 ar given to find a, or c and @ given to 
find 4, we get the result by subtraction, the opera- 
tions in these two cases being of the same nature be- 
cause of the commutativ law for addition. 

The operation of addition is always possible. The 
operation of subtraction is possible only when the 
passiv number is greater than the activ number. 





(1) See Stolz und Gmeiner, p. 5 ; Schubert, Encyklopadie, p. 8. 
(2) See Grassmann, p. 5. 


Sn i 


oo 
= 


Te 





legouncdl JBL 
INTEGERS: 


ChAT EEK att: 
POSITIV INTEGERS. ZERO. NEGATIV INTEGERS. 


151. We now proceed to invent a new system of 
numbers and will, therfor, for the sake of distinction, 
call the numbers hitherto treated of primary numbers (’). 

152. We have seen that, when a> 4, the symbol 
a—bhas a meaning, that is, the two primary num- 
bers a, 6, when written in this order and connected by 
the sign —, determin a certain concept, the difference ° 
between a and 4. When a>} 4, the symbol a — @ is 
meaningless and the numbers a, J determin no con- 
cept, when written in this order and connected by the 
sign —. It will be useful, however, to hav the num- 
bers a, 6 determin an idea even when a > 4, which 
shall hav properties like those of the difference a — 4, 
when a > 4. 

153. Definition. Let us agree, then, that every two 
primary numbers a, 4, taken in this order, shall de- 
termin a concept (’), to be calld an integer, this con- 





(1) See Lefevre, p. 19. 
(2) For this method of defining an integer see Dini, ‘‘ Grundlagen fiir 
eine Theorie der Functionen einer verainderlichen reellen Grésse,’’ trans- 


61 


62 INTEGERS. 


cept to be the difference a — 4, when a> 4(’), that 
is, when the symbol a — 6 has a meaning, and in all 
other cases to hav properties like those of a difference. 
Let us also call this new concept a number. 

Stated briefly, then, an integer is a number couple 
made by combining two primary numbers a, @ in this 
order, the number couple being the difference a — J, 
when a> 4, and in all cases having properties like 
those of a difference. 

Thus every two primary numbers determin an in- 
teger. 

In the case when the integer is a difference it is 
univalent by § 77; let us agree that in all other cases, 
also, the two primary numbers shall determin only 
one integer. 

Then every two primary numbers will determin 
one, and only one, integer. 

This concept, the integer determind by the primary 
numbers a, 4, is not the difference between a and 3, 
except when @> 4, but a new idea created by our 
imagination, a compound idea made up of the two 
simple ideas a and 0. 

154. We might use for this compound idea any 
symbol to which a meaning has not already been 


lated from the Italian by Liiroth und Schepp, 1892, p. 3. The idea of 
‘number couple’’ is due to Wm. Rowan Hamilton. See the 77ams- 
actions of the Royal Irish Academy, 1835, Vol. XVII., part 2, p. 293; 
‘¢Hamilton’s Lectures on Quaternions,’’ 1853, preface, p. 8; Stolz 
und Gmeiner, p. 51. 


(1) See Dini, p. 4; Stolz und Gmeiner, p. 48. 


POSITIV, ZERO, NEGATIV. 63 


attacht and which should show distinctly the two 
primary numbers of which the integer is composed. 
Thus we might use the symbol @— 43 ('), which by 
agreement represents the same idea as the integer 
when a> 40. But we would then be likely to confuse 
the idea of integer with the idea of difference. We 
will therfor use the symbol (a, 0) (’). 

Thus, for example, (5, 2), (4, 4), and (3, 7) ar 
integers. 

155. Definition. The primary numbers a and 6 will 
be calld the elements of the integer (a, 0), a the first 
element and 2 the second. 

156. The order of the elements is essential. Thus 
(a, 0) is not necessarily the same integer as (J, a). In 
case, however, the two elements ar the same, inter- 
changing them does not alter the integer. 

157. For the sake of clearness, we will continue to 
use the Roman letters a, 0, c, d, --- to denote primary 
numbers and will use the Greek letters a, 8,7, 0, --- to 
represent integers. We will also employ one Roman 
letter with the distinguishing subscripts 1, 2 to denote 
the two elements of any given integer, and the corre- 
sponding Greek letter to represent the integer itself. 

Thus a represents the integer (a,, a,), 3 the integer 
(4,, 5,), 7 the integer (c,, c,), and so on. 


(1) See statement of Principle of Continuity, or Permanence, in Pea- 
cock’s ‘* Symbolical Algebra,’’ 1845, p. 2 and in Hankel’s ‘‘ Theorie 
der komplexen Zahlsysteme,’’ 1867, § 3. See also Lefevre, p. 76. 

(2) See Stolz und Gmeiner, p. 47. 


64. INTEGERS. 


158. Theorem. Zhe system of integers includes the 
system of primary numbers. 

Let @ be any given primary number. Choosing a, 
at random and setting a, = a, + a, we hav a, — a, =a. 
Hence the integer (@,, @,) is the primary number 
a, — a, or a, S215 3 

159. Definition. Integers may be divided into three 
classes according as the first element is greater than, 
equal to, or less than the second. 

Integers of the first class are calld positiv, those 
of the second zero, and those of the third negativ (’). 

Thus (2, 1), (5, 3), (9, 5) ar positiv integers, (1, 1), 
(3,53), (7,°7). at zero intesers (ands 234 2,07) nee) 
ar negativ integers. 

160. Theorem. AU positiv integers ar primary num- 
bers, and conversely. 

161. Definition. The integer obtaind from a given 
integer a by interchanging its elements is calld the 
opposit, or negativ, of a and is indicated by the symbol 
a or a, the sign — being written directly over a, or 
above and slightly to the left (*). This symbol is 
univalent (§ 153). Thus, if a is (a, @,), @ is (a, @,). 
Foriexamplemif 15) (8,23). Sst 500,): 

The minus sign,.as here used, is a sign of operation. 

Sometimes, to more clearly distinguish a from @, a 
plus sign is written over a, thus 4, or above and 
slightly to the left, thus ‘a. 





(1) See Dini, p. 4. 
(2) For this symbol see Stolz und Gmeiner, p. 54. 


POSITIV, ZERO, NEGATIV. 65 


Since every primary number is an integer (§ 160), 
every primary number has its opposit. 

162. Theorem. Avery negativ integer ts the opposit 
of some primary number 

Thus, suppose (@,, @,) is negativ. 

Then as < 2, § 159. 

Now (@,, @,) is the opposit of (@,, a,) wher a, > @,. 

Hence (a,, @,) is the opposit of the primary number 
a, — a,, that is, (@,, a,) is the same as ~(a, — @,). 

If a, — a, = @, (@,, @,) is the same as @. 

For example, (4, 6) is 2, (5, 9) is 4, (1, 8) is 7 

163. The signs + and — over primary numbers 
may be regarded as signs of quality showing whether 
the integer thus represented is positiv or negativ. 

164. Theorem. The opposit of a positiv integer is a 
negativ integer, the opposit of a zero integer 18 a Zero 
integer » the opposit of a negativ integer is a positiv 
integer. 

If (2,, 2,) is positiv, a2, > @,, 

Hence a, < a, and (4,, a,) is negativ. 

Similarly the other parts of the theorem may be 
proved. 

165. Theorem. The opposit of the opposit of a given 
integer ts the given integer. 

166. Theorem. // we znterchange the elements of an 
integer an odd number of times, the resulting integer ts 
the opposit of the given integer ; if we interchange the 
elements an even number of times, the hat as the given 
integer. 

ao 


66 INTEGERS. 


167. Let us proceed now to giv integers properties 
analogous to those possest by differences. Let us 
define the terms equal, greater, less, sum, difference, 
etc., in connection with integers. These terms ar at 
present meaningless with reference to integers. We 
can therfor giv any definitions for them that we please, 
provided the definitions ar not inconsistent with each 
other or with preceding definitions. We must remem- 
ber in particular that, when a > @, the integer (a, 0) is 
the difference a — 0, a primary number, and that for 
such integers the above terms ar already defined. 

168. Definition. Equality("). To obtain a proper 
definition for the term “‘equality”’ with reference to 
two integers (a,, @,) and (4,, 4,), let us consider first 
the case when a, > a, andd, > 4,, in which case these 
integers ar the primary numbers a, — a, and 0, — 4, 

By § 126 a necessary and sufficient condition for 
the equality of a,— a, and J, —4,, when a, > a, and 
b, > 6,, is that a, + 6, should equal a4,+0,. Ifwemake 
this condition our requirement for the equality of any 
two integers (a,, a,) and (4,, 4,), ther will be no con- 
tradiction of definitions and the property of equality 
of two integers will be like that of equality of differ- 
ences. Formally then we define: (a@,, @,) = (4,, 4,), if 
a,+6,=a,4+ 0. 

Thus (4 ye=1(0, 80), since 7+6=4-+0; 
(Oy (1, 5 SINCE 5p = ear 


(1) See Dini, p. 3; Davis, ‘‘ Logic of Algebra,’’ 1890, p. 17; Stolz 
und Gmeiner, p. 48. 





a) 


POSITIV, ZERO, NEGATIV. 67 


Pode Meoremine 7 0) 0, ard a 0, | then 
(4, @,) Ok (4, b,). 
Ror, then, a+6,=a, + 4, § 50. 
PO mL neOrenh a. 101 (47, @,)"— (2, 22). 
This follows from § 169, 
since Gea Gea Cae ae SEO; 
177i. Theorem. /7 a= 8, then B = a. 
pieemoe—- Ol (¢,, 7, jo (0), 01), 


a,+6,=a,+ b,. § 168. 
Hence a,+b,=a,+ 4, Se 
Or b,+ 4,= 6,4 4,. § AI. 
Therfor B= a § 168. 


This theorem is the Commutativ Law for Equality 
of Integers. 
172, Theorem. /f a+ 8, then B+ a. 


This is proved in the same way as § 8. 
173. Theorem. /fa=f and B=j7 thena=y. 


- Since Se a,+6,=4a,4 4,. 
pincer s:— 7, b+¢6,=6,4+¢,. 
Adding, (a, 45 b,) ce (2, = é,) a (4, aR b,) 15 (2, = ¢,). 
§ 50. 
Or (2, + &%) + (4, + &) = (@ + 4) + (4 + 4). 
§ 59. 
Hence a+o=a,4+¢,. § 48. 


Therfor (OE. 


68 INTEGERS. 


174. Theorem. /f a=f and By, then a+y7. 

This is proved in the same way as §35. 

175. Theorem. /f we hav a series of equalities in 
which the right member of each equality, except the last, 
7s also the left member of the succeeding equality, as 
a—=Ppa=y=O0=—6 =---, then any intéser im ihe series 
7s equal to any other integer in the series. 

176. Theorem. /f a={§, then a is positiv if B ts 
positiv, a is zero uf B 1s zero, and ats negauv if B ts 
negatv. 


Dice 7. a,+6,=4,4+ b,. 
If f is positiv, oS Ee 


Hence Cee ST be S126; 

Therfor @ is positiv. 

Similarly the other parts of the theorem may be 
proved. 


177. Theorem. /fa=f, then a=f. 

178. Theorem. /f we hav a series of equal positiv 
entegers, they all represent the same primary number. 

SUDDOSe/ d=) 9 == yee 


Then, since a, 8, 7,--- ar all positiv, this means 
that a4, -—@,=6,—6,=q—-cqQ=::: 

That is, a,—4@, 6,—0, c,—¢,+-- ar the same 
primary number. Sa5 

Therfor a, 8, 7, --- all represent the same primary 
number. 


Thus (4, 2), (5, 3), (8, 6) all represent the primary 
number 2. 


POSITIV, ZERO, NEGATIV. 69 


179. Theorem. /f we hav a series of equal negativ 
integers, they ar all opposits of the same primary number. 


Suppose a= P =y =.--and that a, P, 7, --- ar neg- 
ativ. 
Then a=8=y=---and a, f, 7, --- ar positiv. 
oe S851 77,104, 
Hence a, f, 7, --+ all represent the same primary 
number. §178. 


But a, f, 7, --- ar the opposits of a, B, 7,--- § 165. 
Therfor a, 8, y, --- ar all opposits of the same pri- 
mary number. 
Thus (2, 7), (3, 8), (4, 9) ar all opposits of the pri- 
mary number 5 and may be represented by the symbol 
cs 
"180. Theorem. A// zero integers ar equal (’*). 
Let a and £ be any two zero integers. 


endo 2. and 0 —='0,. § 159. 
Hence a,+ 6,= a, + 4,. 

Therfor C=; 

Hence, if a, 8, y,--- ar a series of zero integers, 


181. Definition. We saw in § 178 that a series of 
equal positiv integers all represent the same idea, a 
certain primary number. 

In §§ 179, 180 we saw that a series of equal nega- 
tiv integers ar all opposits of a certain primary num- 
ber and that all zero integers ar equal. 





(1)See Dini, p. 4. 
| > 


70 INTEGERS. 


We will therfor, for the sake of uniformity in our 
system of numbers, say that all equal negativ integers 
ar the same number and that all zero integers ar the 
same number. The latter number we will call zero, 
cipher, or naught and will represent by the symbol 
Oy. 

Thus o stands for (3, 3), (5, 5), (8, 8), «=> 

182. We may therfor henceforth, when any two 
integers ar equal, say they ar the same integer. 

183. Theorem. O=o0. 

184. Theorem. @=<a. 

185. Theorem. J/ the same primary number be added 
to cach element of an integer, the integer 1s unchanged. 

In symbols (a, + 2, a, + &) = (4, @,). 

This follows immediately from the definition of 
equality. 

186. Definition. Greater. Less(*). In order to dis- 
cover a proper definition for the terms “ greater” and 
“less”? with reference to two integers (a,, @,) and 
(2,, 4,), let us consider first the case when these inte- 
gers ar the primary numbers a, — a, and 4, — 4, 

By § 128 a necessary and sufficient condition that 
a, — a, should be greater than 4, — 0,, when a,> a, 
and 6,>4,, is that a,+ 6, should be greater than 
a,+6,. If we make this condition the meaning of the 
statement “(a,, a,) is greater than (4,, 0,),” ther will be 





(1) The symbol o is used in an Indian manuscript of the eighth 
century to denote the absence or lack of number. 
(2) See Davis, p. 17 ; Stolz und Gmeiner, p. 55. 


POSITIV, ZERO, NEGATIV. fia 


no contradiction of definitions. Formally, then, we 
define: (@,, @,) > (4, 4), if a, + 6,> a, + b,. 

Biilatiym wee CctiNne sm (7. 0 77,) <8 (0) s-0:).. af 
atb,<a,t+,. 

The properties of inequality of integers ar, then, like 
those of inequality of differences. 

187. Theorem. Having given any two integers a 
and 3, one of the statements a=P,a> P, anda<P 
must be true, and only one can be true. 


For one of the statements a, + 6, = a, + 4, 
@,+6,>a,+ 06, and a,+6,< a, + 6, must be true, 
and only one can be. § 1209. 

188. Theorem. // a> f, then B <a, and conversely. 

lie 2, a+6,>a,+ 4, § 186. 

Hence a,+6,<a,+ 6, § 89. 

Or b+ a,< 6, +t a,. §§ 41, 117. 

That is, fekees ee § 186. 


For the converse theorem reverse the steps. 

189. Theorem. Jf a>, thena<f. 

190. Theorem. /f a=f and B>7, thena>yj. 

The proof of this ENS is formally the same as 
that of § 173. 

191. Theorem. /fa>f and P>yj7, thena>yj. 

192. Theorem('). a<Pand P=y7, thn a<yjy. 


Siltea—p= 7, == BP > a. §§ 171, 188. 
Therfor ie § 190. 
Or any, § 188. 


(2) See Stolz und Gmeiner, p. 6. 


72 INTEGERS. 


193. Theorem. Jfa<fP and P <y, thena<yj. 
The proof of this theorem is like that of § 192. 
194. Theorem('). /f a>f and B=y7, then a>y. 


For eithera=7y,a<y, ora>7. S187; 
live a= 7 Since als0, 7 e=:0, 00 =e. § 173. 
But this contradicts the hypothesis. § 187. 
Hence ay 
Similarly 26g 
Therfor Coan 


195. Theorem. [fa=P and BP <j, thna<y. 

This follows from § 194 by the method used to 
prove § 192. | 

196. Theorem. J/f we hav a series of inequalities of 
the same kind,asa>B>y>d>e¢>.---, any imteger 
in the series 1s greater than any following integer. 

The proof is the same as that of § 116. 

197. Theorem. /f we hav a series of equalities or 
enequalities of theformaspSysSosSe5...-, any in- 
teger in the series is equal to or greater than any follow- 
ing integer, the sign being >, if ther is at least one 
sign of inequality in the series between the two integers. 

The proof is the same as that of § 117. 

198. Theorem. <Azy given set of integers, no two of 
which ar equal, may be arranged in order of magnitude. 

This theorem is proved in the same way as § 144. 

199. Theorem. Jz any given set of integers, no two 
of which ar equal, ther is one greatest and one least. 


(1) See Stolz und Gmeiner, p. 6. 


POSITIV, ZERO, NEGATIV. PS 


When the given set is arranged in order of magni- 
tude, the last integer is greater than all the others 
($ 196); the first integer is smaller than all the others. 

Ther cannot be two, each of which is greater than 
all the others. ; 

For, suppose A and # were each greater than all the 
others. 

Then 4> yp and p> A. 

Suncewee> A Acs 

But this contradicts § 187. 

Hence ther is only one integer greater than all the 
others. 

That is, ther is one greatest integer. 

Similarly ther is one least integer. 

200. Theorem. very positiv integer 1s greater than 
zero; every negativ integer is less than zero; and con- 
versely, every integer greater than zero ts positiv ; every 
integer less than zero is negativ. 

For, if (a, @) is positiv, a, > a. 


Hence a+6>a,+ 8. SHE 
Therfor (a, a,) > (4, 0). 
Or (ag0d,) a> 0. 


Similarly the second part of the theorem is proved. 

For the converse theorem reverse the steps. 

201. Theorem. Avery positiv integer is greater than 
every negativ integer; every integer greater than a 
posit integer ts positiv ; every integer less than a nega- 
tiv integer 1s negativ, 


74 INTEGERS. 


202. We saw in §§ 5, 75 how to compare two 
positiv integers ; in § 201 we learnd that every positiv 
integer is greater than every negativ integer. The fol- 
lowing theorem will giv a rule for comparing two 
negativ integers. 

Theorem. Jfa>b,a<b; fa=b,a=); fa<i, 
a>b; and conversely, fa <b, a>; a= 0d, abe 
DRESS a<b. 

This follows immediately from §§ 177, 189. 

This theorem might be abbreviated as follows: 


a= b according asa=b. 
— 


203. Definition. Sum(") In order to derive a 
proper definition for the sum of two integers (@,, @,) 
and (%,, 4,), let us consider first the case when these 
integers ar the primary numbers a, — a, and 0, — 4, 

By § 121 the sum of the primary numbers a, — a; 
and 6,— 4, is (a,+ 6,)—(a,+ 4,), which with our 
present notation is represented by (a, + 4,, a, + 4,). 
Moreover the symbol (a,+ 4, @,+ 4,) is univalent 
in all cases, since a, + 6, and a, + 0, ar univalent. 

§§ 40, 153. 

If, then, we make the integer (a, + 4, a, + 0,) the 
sum of (a,, a,) and (4,, 4,) in all cases, ther will be no 
contradiction of definitions and the sum of two inte- 
gers will hav a form similar to that of the sum of two 
differences. . 


(1) See’Dini, p. 3. 


POSITIV, ZERO, NEGATIV. 75 


Formally, then, we define: (@,, a,) + (4,, 4,) repre- 
sents (2, + 0,, a, + 4,). 

204. Operation of Addition of Integers. Since in 
getting the sum of two integers (a,, a,) and (4,, 4,) we 
add a, to 4, and a, to 0,, we may be said to obtain 
that sum by an operation, the operation of adding 
(a,, a,) and (4, 4) 

205. Theorem. (¢,,2,) +(0,, 0,)=(a@, + 4, a, + 4,). 

For, by definition, (4, @,)+ (4, 4,) stands for 
(a, +4, 4, +4). 

And any integer is equal to itself. Sah70; 

206. Theorem. The sum of two positiv integers 1s a 
positiv integer ; the sum of two zero integers 1s a zero 
integer; the sum of two negativ integers 1s a negativ 
integer ; the sum of a positiv and a zero integer ts post- 
tiv; the sum of a negativ and a zero imteger ts nega- 
tiv ; and conversely, tf the sum of two integers ts positiv, 
one of thems positiv ; of the sum is negativ, one 1s nega- 
tv. : 

Suppose a and f ar both positiv. 

Then ae a pan 0, > Oe. § 159. 

Hence a+6,>a,+ 6, Serr 


Therfor a + f is positiv. 

Similarly the other parts of the direct theorem may 
be proved. 

The converse theorem is proved by the method of 
exhaustion. 


207. Theorem. a+ fP=f+4. 


76 INTEGERS. 


For a,+ 6,=6,+a,anda,+6,=6,+ a, § 41. 

This theorem is the Commutativ Law for Addition 
of Integers. 

208. Theorem. a+ a=oO. 

For (@,, @,) + (@,, @,) = (a, + 4, @ + 4,), which equals 
zero. § 1509. 

209. Theorem. a+o= a. 

For (a@,, @,) + (8, 6) = (a, + 4, a, + 4), which equals 
(2a § 185. 

210. Theorem. O+0=0. | 

211. Theorem. “(2+ f)=a@ +B. 

In words: The opposit of the sum of two integers ts 
the sum of thei opposits. 

212. Theorem. /fa=f, then at+y=fP+y7, and 
conversely. 

For the direct theorem, 


since a=f,a,+ 6,=a,+4+ 4. § 168. 
Adding ¢c, + ¢c, to each member of this equality, 

(a, + 4) + (4 +6) = (4, + 4) + ( + &).§ 46. 
Or (a+ ¢) + (6, + ¢) =(@,+ 6) + (6 + 4). §59. 
Therfor (a, + ¢, @ + ¢) = (+6, +6). § 168. 
Or a+y=8+7. PSHzO3, 


For the converse theorem reverse the steps. 
213. Theorem. /fa= ff, theny+a=7+ Ff, and 
conversely. 


POSITIV, ZERO, NEGATIV. Vii, 


This follows from §§ 207, 212, or it may be proved 
independently of §§ 207, 212 by a method similar to 
that used for § 212. 

214. Theorem. /f a= a Gd aml Oe LACH 
a+y=f+4+0; and conversely, faty=Pt+o 
and + = 0, thena= f. 

This is proved in the same way as § 50. 

215. Theorem. /f a>, thena+y>f8+y7, and 
conversely. 

The proof of this theorem is formally the same as 
that of § 212. 

216. Theorem. /f a>f, thany+a>7+4+, and 
conversely. 

2i7, Theorem. /f a> and += 0, then 
aty>fB+0; and conversey, ifatyoPt+e 
and y = 0, thena> f. 

218. Theorem. /f a>fP and y>0, then 
aty>fPt+a. 

219. Theorem. Jf a@=8 and y= 2 then 
a+ySP +9. 

220. Definition. Difference. When (a, a@,) and 
(4,, 6,) ar the primary numbers a, — a, and 4, — 4, re- 
spectivly, and a, — a, > 6, — 6,, their difference is, by 
§ 128, (a, + 4,) — (a, + 4,), which with our present 
notation is represented by (a, + 4,, a, + 4,). More- 
over the symbol (a, + 4,, a, + 4,) is univalent in all 
cases, since a, + 6,and a, + 6, ar univalent. To avoid 
contradictions we will make the integer (a, + 4,, a, + 4,) 


the difference of (@,, @,) and (0,, 4,) in all cases. Thus 


78 INTEGERS. 


the difference of two integers is in form similar to the 
difference of two differences. 

Formally then: (a,, @,) — (4, 4,) represents 
(a, + 4, a, + 4). 

221. Operation of Subtraction of Integers. The dif- 
ference a — f of two integers a and f may be said to 
be obtaind by an operation, the operation of subtract- 
ing £ from a. 

The definition of § 220 removes the restriction that 
in subtraction the minuend must be greater than the 
subtrahend. The operation of subtraction is now pos- 
sible in all cases. 

222, Theorem. (a,, a,) — (4,, 4,)=(a, + 4, @,+ 4). 

223. Theorem. a— 8 =a-+ f. 

224. Definition. For this reason a — f is sometimes 
calld an algebraic sum. 

225. Theorem. a—f=a + f. 

226. Definition. A plus or minus sign directly 
befor an integer and not following another integer is 
at present meaningless. Let us agree that such a 
sign shall be equivalent to a plus or minus sign over 
the integer. 

Thus 4 a+ fis the same as 4 + P ora+; 

—a-+ fis the same as a + P. 

227. Theorem. From §§ 184, 223, 225, 226 it fol- 
lows that whenever we hav the sign + or — over an 
integer and the sign + or — befor it, we may change 
either of these signs, provided the other is also changed 
Jrom + to — or from — to +. 


+? 


POSITIV, ZERO, NEGATIV. 79 


Thus + & may be changed to — @; 
+ @ may be changed to — @; 
— % may be changed to + @; 
— @ may be changed to + 4. 

228. Theorem. a«@—a=o0. 

229. Theorem. a—oO= a, 

230. Theorem. O—a=4a., 


231. Theorem. O—O=O. 


232. Theorem. — (4 —f)=—a+FP. 

233. Theorem. /f a= 8, a—f ws zero, and con- 
versely, 

First, since a= 8, a, + 6,= a4, +4 34.. § 168. 

Hence (a, + 4,, a, + 4,) is zero. § £509. 

That is, a — f is zero. § 220. 


For the converse theorem reverse the steps. 

234, Theorem. /fa>f,a—f is positiv,; and con- 
versely. 

The proof is formally the same as that of § 233. 

235. Theorem. /fa<f,a—f is negativ ; and con- 
versely, 

236. Definition. If a— is equal to the positiv 
integer 7, 4 is said to be 7 greater than #, and [is said 
to be x less than a. 

237. Theorem. /f a= 8, then a—y=B—y, and 
conversely. 


tc 7 aty=f+7. $212) 
Or a—y=fP—-y7. $1224. 


For the converse theorem reverse the steps. 


80 INTEGERS. 


238. Theorem. /f a=, then y —a=y—§, and 
conversely. 

239. Theorem. /f a= and y=0, thena—y=PB—O ; 
and conversely, ifa—y = 8 — dandy =0, thna=B; 
ffa—y=P—d anda=f, theny =O. 

The first part of this theorem is usually stated: 
Lquals subtracted from equals leave equals. 

240. Theorem. (4,, @,) =a, — a, im all cases. 


For a, = (a,4+ 4,0) § 153. 
and Gar (Gy ie). 
Hence a,—a,=((a,+ 4)+ ¢, 6+ (a,4c)) § 230. 
=(4,+ (6+), a+(64+ ¢)) 


S$ 54, 169. 
=< (a, a,). § Tose 
Therfor (a,, a,) = a, — @,, §§ 175, 171. 


241. Theorem. /f a>, then a—y>B—y, and 
conversely. 

The proof of this is formally the same as that of 
§ 237. 

242. Theorem. /f a>, then y—a<y—f, and 
conversely. 

Since af, a< fp. § 189. 


Hence rta<7+f. §§ 188, 216. 


Or y—a<y—f. §§ 223, 197. 


POSITIV, ZERO, NEGATIV. 8I 


243. Theorem. /f a>? and y=0, thena—y >B—0 ; 
and conversely, if 2—y > B—O0 andy =0, then a> B. 

244, Theorem. /fa=f and ;>0, thena—y<f—0; 
and conversely, ff a—y7 <B—0 and a=, theny >. 

245. Theorem. /f a> and y<0, then a—y > B—O.~ 
_ 246. Theorem. /f a> Bandy =0, thena—ySf —0. 
247. Definition. As we connected sums and differ- 
ences of primary numbers by plus and minus signs, 
so we may connect sums and differences of integers by 
plus and minus signs, using parentheses to indicate the 
order in which the various operations of addition and 
subtraction ar performd. Every such expression is 
univalent, since every sum and every difference is 
univalent. §§ 203, 220. 

Such an expression is calld a complex algebraic sum. 
The integers from which it is formd ar calld its elements. 

If the signs between integers in a complex algebraic 
sum ar all plus, the complex algebraic sum is calld 
simply a complex sum. 

248. Theorem. Jf for one or more elements of a 
complex algebraic sum equal integers ar substituted, the 

complex sum 1s unchanged. 


249, Theorem. (2+ 8) +y=a+(8+7). 

For (a+ B) +7 =((a +4) + cy (a+ 4) +4) 
and a+(P+7)=(4,4+ (6,4¢,), a + (6,+¢, )). 

But (4,+4)+4=44+(6,4+4) § 54. 
and (a, + 5,) + ¢, = 4, + (4,4 ¢). 


Therfor (2 + 8) +7 = 2+ (2 +7). 
rte 


® 


WP. + 


$2 INTEGERS. 


This theorem is the Associativ law for Addition of 
Integers. 

250. Theorem. <Any complex sum of integers ts in- 
dependent of the arrangement of its parentheses and the 
order of its elements. 

This is proved in the same way as the corresponding 
theorem for primary numbers, § 59. 

This theorem is the Generalized Associativ and Com- 
mutativ Law for Addition of Integers. 

251. Hence we may speak of a sum 
a+fB+7+6+¢+4-.-- without any parentheses, and 
may write the-elements of the sum, a, £, 7, 0, ¢, ---, in 
any order we please without altering it. 

252. Theorem. Zhe sum of the left members of any 
number of equalities 1s equal to the sum of the right 


members. 
Lhus, ifa = 9 70) .6=6, 70) then 
aty+etnt+---=84+04C04+64... 


This may be proved in the same way as § 57. 

253... Theorem. /f a= Sey 0) o> Gj = 0a 
thnaty+e+yn+---S8+04+ 054064... 

If the sign of one of the given statements is >, the 
sign of the result is >. 

254. Theorem. Zhe sum of any number of positiv 
integers 1s a positiv integer ; the sum of any number of 
zero integers ts a zero integer ; the sum of any number 
of negativ integers 1s a negativ integer. 

To prove the first part of the theorem, we know 
that the sum of two positiv integers is a positiv in- 


POSITIV, ZERO, NEGATIV. 83 


teger (§ 206). Hence the sum of that sum and a 
positiv integer is positiv, and so on. By applying the 
' rule z— 1 times we prove that the sum of z positiv 
integers is a positiv integer. 
Similarly the other parts of the theorem ar proved. 
205. Theorem. he sum of any number of zeros ts 
equal to zero. 


256. Theorem. —(a+8+---+0+7+04) 

: =a+h+-.-+F4+74 0. 
For —(a+8+---+0+7+ 9) 
SOs ete tec ait 0} 
—((@+8+4+---)+o +) 49 
—[(@+8+--)+o +740 


E= ec, 


| 


| 


257. Consider next any complex algebraic sum. 
Change all the — signs befor integers and _ befor 
parentheses to +, at the same time changing the 
signs over the corresponding integers or parentheses 
(§227). Next remove all — signs from over paren- 
theses, changing the signs over all the integers in 
these parentheses (§ 256). 

Take the following expression as an example : 


Beer (7 — 0)+(°—C)) + 9) — 9] + (8 + A)f. 
Making the above mentiond changes, this be- 
comes 


{[a+ (8+ (7 +9) ++ 2)) +4) + 0) + O+ A}. 


Sten, 


84 INTEGERS. 


We now hav an expression in which all the signs 
connecting integers ar + signs. All parentheses may 
therfor be removed (§ 251). Thus we get 


Z+BtytO+E+E+ Ft G+ ore 


The sign over any given integer in the result can be 
obtaind by counting the minus signs which ar over 
the integer, befor it, and befor the parentheses within 
which it lies in the original expression. These ar 
calld the minus signs that affect the given integer. 

If the number of these minus signs is odd, the sign 
over the integer in the result will be minus; if their 
number is even, the sign over the integer will be plus, 
or ther will be no sign over it (§ 166). For, every 
time a minus sign is removed from befor an integer, 
the sign over the integer must be changed; and the 
same must be done when a minus sign is removed 
from in front of a parenthesis in which the integer lies. 

Thus in the given expression ther ar four minus 
signs affecting 0, one over it, one befor it, and two in 
front of parentheses within which it lies. Therfor the 
sign of d in the result is plus. Ther ar three minus 
signs affecting ¢«. Hence the sign of ¢ in the result 
~ is minus. 

The principle proved above may be stated as 
follows : 

Theorem. Having given any complex algebraic sum, 
we may remove all parentheses, if we write a plus sign 
befor every integer and place over the integer a minus or 


POSITIV, ZERO, NEGATIV. 85 


a plus sign, according as the number of minus signs af- 
fecting it 1s odd or even. 

This theorem may be calld the Associativ Law for 
Addition and Subtraction of Integers. 

258. Definition. The expression considerd above 
may be written without any signs over its integers as 
follows: 


{(({(((— «+ 8) +7] + 9) -— 8} 
—€)—7] + 9-4} -8£. 


Such an expression is calld a standard algebraic 


SS 2508227: 


sum. 

The parentheses ar usually omitted from a standard 
algebraic sum. Thus the above expression may be 
written —a+8+y7+0—e—C€—7+0—0-f. 

In such an expression the operations ar supposed 
to be performd from left to right ("). A parenthesis 
could not be inserted as follows: 


meet ne aCe) apt 1d B. 


In what follows we will use the term algebraic sum 
to indicate a standard algebraic sum whose parentheses 
hav been removed. 

259. The integers in the algebraic sum under con- 
sideration in § 257 may also be arranged in any order 


§ 250. 


(7) See E. Bardey, ‘‘ Resultaten zur Aufgabensammlung,’’ 1871, p.2; 


Schréder, ‘‘ Lehrbuch der Arithmetik und Algebra,’’ 1873, Vol. I, p. 
219; Stolz und Gmeiner, p. 9 


y 


Yb 


86 INTEGERS. 


Thusat+P+y+o+tet+6+7+64 9+ 8 may 
be written 9 4 0a ty oe ete ipa mg ctnor 
or Btéd0—at+t+y—e—C+0—7—f-8. 


We may state this result as follows: 

Theorem. Having given any algebraic sum, the 
order of its elements may be changed in any way, pro- 
vided the sign befor cach element goes with tt. 

This may be calld the Commutativ Law for Addi- 
tion and Subtraction of Integers. 

260. A convenient arrangement of the elements in 
any algebraic sum is one in which all those preceded 
by the plus sign ar written first and then those pre- 
ceded by the minus sign. 

Thus the expression considerd above would be- 
come 8+7+0+0—a—¢e—€—7y—d-f. 

The integers preceded by the minus sign may now 
be enclosed in a parenthesis with plus signs between 
them and a minus sign befor the parenthesis (§§ 223, 
256). Thus the above expression is equal to 


B+yto+0—(atet+C+y+0+ 8). 


This result may be stated in the following theorem. 

Theorem. Axy algebraic sum is equal to the sum of 
those of its elements which are preceded by the plus sign 
minus the sum of those which ar preceded by the 
MINUS SIEN. 

261. Cancellation. Suppose we hav an algebraic 


“ sum two of whose elements ar equal, but one preceded 


POSITIV, ZERO, NEGATIV. 87 


by the plus sign and the other by the minus sign. 
Then these integers with their signs may be removed 
from the given expression. 

For, suppose the equal elements with their signs ar 
+aand—{. The order of the elements of our sum 
may be changed so as to bring these integers to the 
end of the expression. Represent all of the algebraic 
sum thus obtaind except the +a and —f by vp. 
The algebraic sumis then reduced tothe form u +a — Pf. 
This may now be written » + (a — ), or uw + 0(§ 233), 
which equals yp. 

Thus in the algebraic sum considerd in § 260, 
since B =f, BP and — 8 may be omitted ; similarly 
+06 and —d may be omitted. This may be indicated 
by drawing a line thru these letters, or cancelling (’) 
them, thus: 


Reg 6a Ca 8 aR. 


The given expression is, then, equal to 


Fe aaa Nea, 


We may state the result proved above as follows. 

Theorem. Jf zz any algebraic sum two elements ar 
equal, but one ts preceded by the plus sign and the other 
by the minus sign, these elements with their signs may 
be removed from the given expression. 


(1) See Fisher and Schwatt’s ‘‘ Text-Book of Algebra,’’ 1898, Part 
Lp. 25. 


88 INTEGERS. 


TRANSPOSITION. 
262. Theorem. /fa=f, thena—B=0. 


Since a = f, a—Bp=P—f. Raz372 
Or a—PB=o. § 228. 
263. Theorem. /fa= — 8, thena+fB=o0. 
For a—(—fP)=a+f. § 225. 
264. Theorem. /fa=f8+47, thena—y=f. 
Sincea=B+y7, a—y=8+7—y7. § 237. 
Or a—y=f. § 261. 


265. Theorem. /fa=P—y7, thena+y=f. 

266. Theorem. <Axy znteger in either member of an 
equality may be transposed (') to the other, of its sign ts 
changed, the member from which the integer 1s removed 
becoming sero, of the integer transposed was the entire 
member. 

This follows immediately from §§ 262-265. 

267. Theorem. Zhe szgus of all the integers in an 
equality may be changed. 

This is proved by transposing all the integers, by 
§ 266, and then reversing the equality. 


Thus, if at+Pp=—7+4+06, 
y—d0=—a—Ff. 
Hence —a—Pp=7—0. 


(1) See Beman and Smith, ‘‘ Elements of Algebra,’’ 1900, p. 148. 


POSITIV, ZERO, NEGATIV. 89 


268. Theorem. /fa>, thena—pP>o. 

This is proved in the same way as § 262. 

269. Theorem. /fa> —f, thena+ P>o. 

270. Theorem. /fa>f8+y7, thena—y>Ff. 

271. Theorem. /fa>P—y, thena+y>f. 

272. Theorem. /fa<f, thena—fB<o. 

273. Theorem. /fa< —f, thena+tfP<o. 

274. Theorem. /fa<f+y7, thna—y<f. 

275. Theorem. /fa<B—j7, thena+y<f. 

276. Theorem. <Axy integer in either member of an 
inequality may be transposed to the other, if the sign of 
the integer 1s changed, the member from which the integer 
is removed becoming zero, f the integer transposed was 
the entire member. 

277. Theorem. The signs of all the integers in an 
inequality may be changed, if the sign of tnequality ts 
changed, from > to < or from < to>. 


Thus, if a—Bo—r—J, 
yto>—at+P. 
Hence —a+tPp<rt+o. 


278. Theorem. Laual integers, one in each member 
of an equality or inequality and preceded by the same 
sign, may be canceld. 

This follows from §§ 266, 276, 261. 

279. Integers orderd. We will now arrange our 
system of integers in a series so that each shall be one 
less than the succeeding one. We hav seen that in 
the natural series of primary numbers each number is 


gO INTEGERS. 


one less than the immediately succeeding number 
($ 133). Starting with any given primary number we 
could therfor, by successivly subtracting ones from it, 
obtain all the primary numbers below it, down to one. 
But we could not continue beyond this point as ther 
is no primary number smaller than one. 

With integers, however, we can continue this process 
as far as we please. 


Thus I—I=0,0O—I=I, 


I—1I=—(1+1)=2,2—1=—(2+ 1) =3,--- 


Thus we hav the series 


“+59, 8, 7; 6, 5; 4, a 25 i O, I, 2, 3,4, ROSS 8, OF as 
If a is any negativ integer in this series, the one 
preceding it isa@— 1or—(a+1). Hence the opposit 
of every primary number will be found somewher in 
the series ($65). And since every negativ integer is 
the opposit of some primary number, every negativ 
integer will be found in the series. We also hav zero 
and every positiv integer in the series. Therfor the 
series includes all the integers (’). 

280. Definition. The series given above may be 
calld the natural series of integers. The number 0 is 
calld the center of the series. The other integers may 
(1) Negativ integers and zero seem to hav been freely used as num- 
bers first by Descartes in the seventeenth century. See his ‘‘ La Géo- 
métrie,’’ 1637. ‘The first traces of such use ar, however, found in the 


work on astronomy of Bhaskara, an Indian mathematician of the 
twelfth century. 


POSITIV,; ZERO, NEGATIV. gI 


be obtaind by adding to and subtracting from zero 
in succession the natural series of primary numbers. 
§§ 209, 230. 

281. Theorem. x the natural sertes of integers each 
integer ts one greater than the preceding integer. 

For, ifa=P—1,P—a=1. 

282. Theorem. Ther is no greatest integer and ther 
7s no smallest integer. 

The first part of the theorem follows from the fact 
that every positiv integer is a primary number and is 
ereater than zero and greater than everynegativ integer. 

- The second part follows from the fact that every 
negativ integer is less than zero and less than every 
positiv integer, and that — (a+ 1)<—a. § 235. 

This theorem is equivalent to the statement that 
the series of integers has no beginning and no end , it ts 
limitless, or infinit, in both directions. 

283. Theorem. J/f a zs any integer in the natural 
series, the integers that immediately precede a ar tn order 
a—I,4—2,a—3,---,a—n,a—(u+12),---,and those 
that immediately follow a ar in order a+T1, a4 2, 
A+ 3,---,a+nua+(n+1),---, wher 1, 2,3, ---, 
n,n4+. I, --- ts the natural series of numbers. 

Let the integers that immediately precede a in order 


be £, if OMe, K, A, +> 


Then 8 = a—I, $2770; 
y=P-—1=(4—1)—-I=a—(1+1)=a—2, 
§ 260. 


d=y7y—1=(a—2)—1=a—(2+4 1) =a—-3, 


Cree Ae a eee to see le <6 so. es eo © & SF el 18) C.F © © . € 8 @ 


92 INTEGERS. 


Moreover, if « = a — 2, 
A=K—-1 =(a—2)—1=a—(n+1). 


Hence the integers f, 7, 0,---, *, A,--- ar equal re- 
spectivly to the integers a— 1, a—2, a— 3, --- 
“—n, a—(#+1),-:: 

In this series the numbers subtracted from a ar each, 
after the first, obtaind from the preceding by adding 1 
to it. These numbers ar therfor the natural series of 
numbers. 


) 


The last part of the theorem is proved in the same 
way as § 69. 

Thus the natural series of integers may be written 
with a as a center as follows: 


,4—(%+ 1) a—4N,.--,a—3,4—2,a—1,4, 
ET, O25 CS ha Maite 2 ee 


284. Theorem. /fa<fP<y, B zs one of the num- 
bers in the natural series of integers between a and 7. 


Let S—a=o0 andy—fB=e. 

Then 8 = a+ dandy; = + ¢, wher 6 and ¢ ar posi- 
tiv. §§ 266, 234. 

The proof is now the same as that of § 139. 

285. Definition. When a</ <7, f is said to lie 
between « and y in magnitude. 

286. Theorem. /f a and f ar any two consecutiv in- 
tegers in the natural series of integers, ther exists no 
integer between a and P. 

Suppose a<7<f. 


POSITIV, ZERO,’ NEGATIV. 93 


Then B—a=f-—y+y—a4 § 261. 
= (6—7)+ 7 — 2), ES257, 
wher 8 —7 and y — @ ar positiv. § 234. 


Therfor 8 —a>fP—;7S1. 

The proof is now the same as that of §141. 

287. Definition. In the natural series of integers 
the alternate integers commencing with zero and going 
either way ar even; the others ar odd. This agrees 
with the definition of § 68 for primary numbers. 

Thus 

oe, 88.042 026 4-0,.8,4--.ar even; 


SF 7> 55 my 2 Us er 5, 7» 9,:-- ar odd. 


288. Theorem. Jf a zs even, a is even; tf ats odd 
a is odd. 

289. Theorem. The sum of two even integers, or of 
two odd integers, 1s even; the sum of two integers, one 
of which ts even and the other odd, 1s odd; and con- 
versely, if the sum of two integers is even, they ar either 
both even or both odd ; of the sum of two integers ts odd, 
one must be even and the other odd. 

For the direct theorem, suppose first that a is an 
even integer. 

Now, if 8 is zero, a+ P=a and a+ will be 
even. § 182. 

If 8 is positiv and equal to J, consider the portion 
of the natural series of integers 


GP, &- 2,4 + 3,-+7, a+ 0, 


94 INTEGERS. 


Since @ is even, the first of these numbers is odd. 
Hence the second is even, the third odd, etc. 
If we make these numbers correspond to the num- 


bers 
I, 2, By eas b, 


we see that the numbers of the first series which cor- 
respond to odd numbers in the second ar odd and 
those that correspond to even numbers in the second 
ar even. 

Therfor a + £ is odd or even, according as f is odd 
or even. 

If 8 is negativ and equal to — 4, we consider the 
series of integers 


a—1,a—2,a—3,::-,a—b6 


and see in a similar manner that a + §, or a — 0, is odd 
or even according as f is odd or even. 

Therfor, in all cases, if a is even, a + f is odd or 
even according as / is odd or even. 

The case when a is odd is proved similarly. 

Each part of the converse theorem may be proved 
by the method of exhaustion. 

For the first part, when the sum is even, ther ar 
only three possible cases. 

Either one integer is even and the other odd, or 
both ar even, or both ar odd. 

If one were even and the other odd, the sum would 
be odd, by the direct theorem. 

But, by the hypothesis, the sum is not odd. 


POSITIV, ZERO, NEGATIV. gs 


Hence we cannot hav one even and the other odd. 

Therfor they ar either both even or both odd. 

The syllogism for the last step of this proof is an 
amplified form of syllogism V of § 131. It may be 
written schematically as follows: 

Wajor Lremis: ither:A is 4, or Cis), ot £ is -F. 

Minor Premis. A is not B. | 

Conclusion = Vnertor either: OC is* D2 or £ isi 

For the second part of the converse theorem, the 
same three cases ar to be considerd. 

We may show by the indirect method that the last 
two ar impossible. 

Therfor the first case must be true. 

The syllogism used here may be written, changing 
the order of the cases: 

Major Prems. Either A is £, or Cis D, or Eis F. 

Minor Prems. A is not B and Cis not D. 

Conclusion. Therfor £ is F. 

290. Theorem. Zhe aiffercuce of two even integers, 
or of two odd integers, 1s even; the difference of two in- 
tegers, one of which ts even and the other odd, ts odd ; 
and conversely, of the difference of two integers ts even, 
they ar either both even or both odd; of the difference ts 
odd, one must be even and the other odd. 

This may be proved indirectly using § 265. 

291. Theorem. Zhe sum of any number of even in- 
tegers 1s even, the sum of an even number of odd tnte- 
gers 1s even; the sum of an odd number of odd integers 
7s odd, 


96 INTEGERS. 


The first part of the theorem follows from § 289 by 
the method used to prove § 254. 

For the second part, if we add the odd integers to- 
gether in pairs, the original sum will be the same as 
the sum of a number of even integers ($ 289). Hence 
this part of the theorem follows from the first part. 

To prove the last part, the sum of all the integers 
except one will be the sum of an even number of odd 
integers, and hence even. Hence this sum plus the 
remaining odd integer will be odd. § 289. 

292. Theorem. Ax algebraic sum of any number of 
even integers 1s even; an algebraic sum of an even num- 
ber of odd integers ts even; an algebraic sum of an odd 
number of odd integers is odd. 

This is proved by changing all the minus signs be- 
for integers in the algebraic sum to plus, changing also 
the signs over these integers, and then applying 
§§ 288, 201. 

293. Theorem. Jf a and f ar two success even in- 
tegers or two successiv odd integers in the natural series 
of integers, B= a+ 2; andconversely,if B=a+2 and 
a 1s even, 8 ts even, if ats odd, f ts odd. 

For the first part of the theorem, 8 =a+4+1+1. 

Hence the theorem follows. 

The converse theorem is proved by reversing the 
steps. 

294. Theorem. Avery even integer can be obtaind 
Srom every other even integer and every odd integer can 
be obtaind from every other odd integer by adding to, 


77 


POSITIV, ZERO, NEGATIV. 97 


or subtracting from, tt a certain number of twos; and 
conversely, every integer that can be obtaind from 
another integer by adding to, or subtracting from, it a 
number of twos 1s even or oda according as that integer 
2s even or odd, 

For, if « and a ar two even integers and « comes 
somewher after a in the natural series of integers, the 
even integer next after ais a4 2 (§ 293), the next 
Plemeee seer tne ne xte@ 142 121-2) and, ‘SO, O11. 
And x is one of this series of numbers. 

If « comes somewher befor a, the even integer next 
befor a is a — 2, the one befor it a — 2 — 2, the next 
a— 2— 2 — 2, and so on. 

Similarly if « and a ar odd integers. 

The converse theorem is proved by applying § 293. 

295. Definition. The system of integers is callda 
closed system with reference to the operations of ad- 
dition and subtraction, because each of these opera- 
tions can be performd on any two integers and the re- 
sult of the operation is always a number of the system. 


CHAPTER Va 
MULTIPLICATION. 


296. Definition. Product. By the generalized as- 
sociativ law for addition of integers the sum of a 
group of z a’s, wher a is any integer and z a primary 
number, is the same, however we group the a’s in 
parentheses in finding their sum. This sum, then, is 
univalent and is determind by a and z('). We will 
therfor call it the product () of a and z. 

This definition implies that ~>1. For to hav a 
sum we must hav at least two elements. But just as 
we speak of a group consisting of only one object, so 
we will speak of adding a group of z a’s, even when 
un = I, the result of the addition being a itself. 

Thus the product of a@ and z may in all cases be 

defined as the sum of a group of x a’s. 
ee the product of « and z may be indicated by either 
of the symbols a x 2, a-x (°), or simply az (‘), 





(1) See Stolz und Gmeiner, p. 21. 

(2) See Schubert, Encyklopadie, p. 14. 

(*) See Euler, ’p. 9. 

(4) The sign X to indicate a product was apparently first used by 
Wm. Oughtred in 1631. See his ‘‘ Clavis Mathematica Denuo Limata,”’ 
1648, p. 10. The sign- was used by T. Harriot in 1631 to denote a 
product. See his ‘‘ Artis Analyticze Praxis,’’? 1631. Stifel in 1544 in 
his ‘‘ Arithmetica Integra’’ indicated the same by juxtaposition. 


98 


te 


MULTIPLICATION. 99 


which ar read “a times z,” or simply “az.” Thus 
an stands fora+a-+a- .-- (# times). 

The integer a is calld the base of the product, the 
primary number z the coefficient. 

The product az is also callda multiple, an n-multiple, 
of a; ax 1 is the unit multiple of a. 

297. Definition. Operation of Multiplication. The 
product az is obtaind from a and z by an operation, 
the operation of adding z a’s. This operation is calld 
multiplication and « is said to be multiplied by z. In 
this operation a is the operand, passiv element, or 
multiplicand, and x the operator, activ element, or 
multiplier ('). 

298. Theorem. Zhe product an is positiv tf a ts 
positiv, zero uf ats zero, negativ if ats nega. 

This follows immediately from § 254. 

209. Theorem.* ax I =a. 

Hole <) t stands for a. “Hence «x Ii==a: 

SalZQ, 

300. Theorem. 1 x I=I. 

301. Theorem. ax x=a+a+ta+.--(x times). 

Fora x a stands fora+a+a- -.-(z times). 

HenceaXu=a+a+a+--. (x times). § 170. 

302. Theorem. I x 2=2. 

This follows immediately from §§ 301, 63. 


303. Theorem. Ox =O. 
This follows immediately from §§ 301, 255. 


304, Theorem. (— a)” = — (an). 








(1) See Schubert, Encyklopadie, p. 14. 


100 INTEGERS. 


Fora +a+ta-+t..-n times = 
—(ata+a++.-.n times). § 256. 


That is, (—a)n= — (an). § 175. 


305. Theorem. J// ezther a or n ts even, an ts even; 
of both a and n ar odd, un ts odd. 

This follows immediately from §§ 301, 291. 

306. Theorem. /fa=f, an = Pn. 

This is proved by adding together z equalities of 
the form a= f. $9252, 016175. 

307. Theorem. /fa> 8, anx> fx. 

This is proved by adding together z inequalities of 
the form a> f. § 253. 

308. Theorem. /f m= 7, am= an. 

For am represents the sum of m a’s and az the 
sum of z a’s. 

Since m =, ther ar as many a’s in the one group 
as in the other. § 5. 

Hence the expressions a and ax ar different sym- 
bols for the same sum. 

Therfor am = an. S170; 

309. As a product is an integer, we may multiply 
it by a primary number or we may combine it with 
other products or integers by plus or minus signs, 
using parentheses to indicate the order in which the 
various operations ar performd. ‘Thus we may hav 
expressions like (a7) + (8m) and a((Z+ m) x). 

All such expressions ar univalent. For every sum, 
difference, and product is univalent. §§ 203, 220, 296. 


MULTIPLICATION, IOI 


Generally, in complex expressions involving multi- 
plication with addition or subtraction, or both addition 
and subtraction, parentheses ar omitted from around 
products("). They should never be omitted from 
around sums or differences, when such omission would 
cause misunderstanding. Thus a/ + mm is understood 
to mean (a/) + (mz) and not a[(2 + m)r]. ) 

310. Theorem. am + an = o(m + n)(’). 

Foram+an=(a+a+a-+---m times) 


+(a+a+a-+---n times) §§ 301, 214. 
=ata+a+---(m+n) times §§$ 250, 37. 
=a(m-+n). | § 301. 

Therfor | 
am + an = a(m-+ n). § 175. 


This theorem is the Left-handed Distributiv (*) Law 
for Multiplication and Addition. 
311. Theorem. ax + fu=(a-+ f)n. 
For an + fu=(a+a+a+.---x times) 
+(P+P+ +--+ z times) 


=(a+/)+(4+$)+(4+/)+--- x times 
= («+ Ay 
Therfor az + Pu = (a-+ B)n. 


“ (2) See foot-note p. 85. 

(?) See Schubert, Encyklopadie, p. 14; Tannery, p. 57. 

(3) The term ‘‘ distributiv,’’ in this sense, was first used by Servois, 
Gergonne’s Annales, Vol. V, 1814, p. 98. 


102 INTEGERS. 


This theorem is the Right-handed Distributiv Law 
for Multiplication and Addition. 

312. Theorem. /f 72> x, om—an=a(m — 1), and, 
fats positiv, am>an,; fats negativ, am < an. 


Since wz>n,m=x+nandm—n= xX. 


Hence am = a(x + 2). § 308. 
Or am=axr-+ anand am—an=ax. 8§ 310, 266. 
Hence am — an = a(m—n). §§ 308, 173. 
Moreover, if a is positiv, ax is positiv. § 298. 


Hence am — an is positiv and am> an. §§ 176, 234. 
Similarly, if a is negativ, am < an. 


313. Definition. According to the definition for a 
product given in § 296, the multiplicand may be any 
integer, but the multiplier must be a primary number. 
We ar now ready to extend the definition to the case 
when the multiplier is any integer. 

By § 312, when 6,>4,, the product a(d, — ,) is 
ab, —ab,; that is, when 4,> 4, a(d,, 0,) represents 
ab, — ab, (§ 240). Moreover the symbol ad, — ad, is 
univalent in all cases. §§ 296, 220. 

If, therfor, we make a/,—ad, the product of @ and 
(4,, 4,) in all cases, our definitions will not be contra- 
dictory. 

Formally, then, we define: a(d,, 0,) represents 
ab, — ab,(’). 





(1) See Schubert, Encyklopadie, p. 15. 


MULTIPLICATION. : 103 


314. Operation of Multiplication of Integers. 

The product af of two integers is obtaind by an 
operation. For ad, and aé, ar each obtaind by an op- 
eration (§$ 297) and then a, is subtracted from a@é,. 
This operation is calld multiplication of integers; a 
is the multiplicand and f the multiplier. 

315. Theorem. af = ab, — ab,, 

For af represents ad, — a4, and hence 


af = ab, — ab,, § 170. 

316. Theorem. I xX a= 4a. 
For 18 ee en DON mAh re Tee S OG, 

=a, —4, &§ 302,230, 

Se, § 240. 
Therfor Ix a=a@. § 175. 
317. Theorem. 0 xX a4=0O. 
For Oxa@=O0X4,—O0X4, 

=0O—O 

ZG, 


318. Theorem. a x O=—O. 


For ao == om Oe. WiCl Cc, = ¢,. § 181. 
SGgrca=—"v., al, = Ol, § 308. 
Hence ac, — ac, = O. $9262: 
Therfor aX O=O. S273. 


319. Theorem. Ox 0 = 0, 


104 INTEGERS. 


320. Theorem. The product of tivo positiv, or of tivo 
negativ integers, 1s positiv ; the product of two integers, 
one of which ts zero, is zero; the product of two integers, 
of which one is positiv and the other negativ, is negativ, 
and conversely, of the product of two integers ts positiv, 
they ar either both positiv or bothnegatv ; tf the product 
7s sero, one of them must be zero; if the product ts nega- 
tiv, one must be positiv and the other negativ. 

Suppose, first, that a and # ar both positiv. 


Then por Wik § 159. 
Hence ab, > ab,. Sail 2: 
Therfor ab, — ab, is positiv. S234. 
That is, a? is positiv. $$:.315, 770, 


The other cases of the direct theorem ar proved in 
a similar way. 
The converse theorem is proved by the method of 


exhaustion. 

321. Theorem. (— a)? = — (a/). 

For (— 4) =(— 4)d,—(— a), pce as 
= — (ab,)— (—(ad,)) $§ 304, 239. 
= — (ab, — ab,) $232. 
= — (4). §§ 315, 177. 

Therfor (— a)? = ~ (a8). S175. 


322. Since the expressions (— a)@ and — (af) ar 
equal, the parentheses may be omitted and both be 
written — af. 


+f 


MULTIPLICATION. 105 


323. Theorem. a(— ?) = — (af). 
For a( — 8) = a(4,, 0,) = ab, — ad, 
= —(ab, — ab,). 
324, Theorem. (— a) (— f) = af. 
For (— a) (— 8) = — («(—8)) § 321, 
short: Cabal $§ 323, 177. 
= ap. § 184 
325. Theorem. (— 1) = — f. 


3826. From the theorems of §§ 170, 321, 323, 324 
we derive the following equalities : 


(+ a)(+ A) = + 48, 

cs a)(+ 8) oe ey aB, 

(+ 4)(— 8) = — a8, 
| (— a)(— 8) = + 48. 

These equalities may be included in the following 
statement. 

Theorem. Jf a product consists of two factors each 
of which 1s an integer preceded by a plus or a minus 
sign, the product is equal to the product of the given 
integers preceded by a plus sign, if the signs of the two 
integers ar alike, but by a minus sign, if those signs ar 
unltke, 

This theorem is the Rule of Signs for Multiplication. 

It is usually briefly stated: Ju multiplication hke 
signs giv +, unlike signs —. 


106 INTEGERS. 


In particular we hav (— @)6 = a(— 6) = — (ad) 


and (— a) (— 6) = ad. 


3827. Theorem. Jf y zs zero, ay = Py. | 


For then ay and fy ar both zero. 
328, Theorem. //y zs zero, ya = 7f. 
329. Theorem. //a— (8) ap == a7. 


Since a = Bf, Come ice 
and Clee 
Hence ac, — ac, = Pc, — e,. 
That is, ee as egg 

330. Theorem. /f/a=f, ya=7f. 
Since a= 8, a,+6,=a, + 3b, 
Hence 7 (4, + 4,) = 7(@, + 4,). 
Or 7a, + 7b, =a, + 70. 
(Pieroram ya, — 7a, = 7b, — 7b, 
That is, escaige! 


§ 320. 


§ 306. 


§ 239. 


§ 315. 


§ 168. 
§ 308. 
§ 310. 
§ 260. 


331. Theorem. /f a= f and y =0, then ay = Bo. 


This is proved in the same way as § 214. 
This theorem is usually briefly stated: EAguals mul- — 


tiplied by equals giv equals. 


332. Theorem. Jf a> and x ts positiv, ay > By. 


Since 7 is positiv, it is a primary number c. 


Sincela 14, ine Ne. 


But, Since }=='cyauaa = 00 


§ 307. 


Ff 


MULTIPLICATION. 107 


and Bc = Py. § 330. 
Therfor Cys § 197. 


333. Theorem. /f/a>1, aa>a. 

This follows immediately from § 332 by putting I 
instead of # and a instead of y. 

334. Theorem. /f » zs any positiv integer, ther ts a 
positiv integer a. such that aa> v. 

If »y > 1, this theorem follows immediately from the 
preceding. 

Riera a ine WE av 2 <a YS 3 3 3c 
meulence, Since 2->il,.weshav 2.x 2 >> v. 

335. Theorem. Tez end. ¥ 1S“ postiiv, ya >> 7B. 


ICC, oy a et Ue a, + O,. § 186. 
Hence 7(@, + 4,)>7(@, + 4,). Sea 2: 
Or 72, + 76, > ya, + 7A,. § 310. 
Therfor 74, — 74, > 70, — 7b,. § 276. 
That is, rgusor> ggee Ses 5: 
336. Theorem. /f a> and jy is negativ, ay < fry. 
Since 7 is negativ, — 7 is positiv. § 164. 
ietee. since 2 > f, 

a(—7)>B(—7). S 332. 
Or — (a7) > — (67). § 323. 
Therfor Oy <ay § 189. 


307. Theorem. Jf a> and jy ts negatv, ya <7. 


108 ; INTEGERS, 


338. From the theorems of §§ 332, 335-337 we 
deduce the following : 

Theorem. /fa+fPandy+o0, ay + fy. 

For then either a> Por ‘a <= 8, “andy as either 
positiv or negativ. 

Hence either ay > fy or ay < fy. 

339. Theorem. Hence, 2f ay = fy, either a=f or 
y= 0. 

340. Theorem. /fa+ Pandy+o0, ya+ 7. 

341. Theorem. /f ya =7f, eithery = 0 ora=f. 

342, Theorem. /fa+t-oandP+o0, aB+o. 


This is proved from § 338 by replacing the # and y 


of that theorem by o and f respectivly. 

3843. From this we may deduce the following im- 
portant theorem. 

Theorem. /f a8 = 0, then either a=o0 or B=O0O. 

344, Theorem. From the last theorem and §§ 317, 
318 we conclude that the product of two integers equals 
zero when, and only when, one of these integers is equal 


lO Zero. 
345. Theorem. Jf ay > Py and ¢ ts positiv, a> Pp. 
For either¢ — 8) or oe soe: § 187, 


If a were equal to f, a7 would equal fy. § 320. 

But ay is given greater than fy and hence cannot 
equal fy. § 187. 

Therfor a cannot equal f. 

Similarly it may be shown that a cannot be less 
than f. 

Therfor a>. 


MULTIPLICATION. 10g 


346, Theorem. /f ay > fy and ; ts negativ, a < Bp. 

347. Theorem. Jf ya>7 and x ts positiv, a> Bp. 

348. Theorem. /f ya>7f and 7 1s negativ, a < B. 

349. Theorem. Jf a> and y=0, then, if x is 
positiv, oy > BO; yx is negativ, ay < B86. 

This is proved like § 217. 

350. Theorem. Jf a> Pandy>0,; then, if a and 
0, or 8 and x, ar both positiv, ar > BO; uf a and Od, or 
B and 7, ar both negativ, ar < Bod. 

If a and o ar both positiv, since 7 > 0, 


ay > ao. § 335. 
Since a> f, ad > Bo. § 332. 
Therfor ay > Bo. § IgI. 


Similarly the other parts of the theorem ar proved. 

351. Theorem. /f a + 8 and 7 = 0, then ay + Bo, 
provided y + O. 

For, if a>, aro according as 7 is positiv or 
_ negativ. 

Similarly if a < P. 

Therfor in both cases ay + f0. 

352, Theorem. /f ay = 0 and 7 =), then a= f, 
provided 7 + O. 

353. Theorem. /f ay = £0 and a>f, then, f a 
and 0, or 8 and y, ar both positiv, or both negativ,; <0. 

This may be proved by the method of exhaustion. 

304. Theorem. Jf ay > f0 anda=§, then, of ats 
positiv, y>0; fats negatv, 7 <4. 


110 INTEGERS. 


355, Theorem. /f ay > 0 and a<f; then, fa 
and 0, or 8 and y+, ar both positiv, > 0. 

356. Theorem. /f ay > $0 and a>; then, if a 
and 0, or 8 and y, ar both negatv, 7 <0. 

357. Theorem. /f aa> fy and a is positiv, either 
a>P or a>yr; f aa> yr and ais negativ, either 


CNB OF a ae 
For the first part of the theorem, either a> f, or 
imate ECP LERC RES 
In the first case the theorem is true. 
iitai— steno § 354. 
If a< f, then ? must also be positiv. § 201. 
Plencen ei S555. 


The second part of the theorem is proved similarly. 

358. Theorem. /f a8 >yy and a is positiv, either 
a>y or P>r; of aB>77 and « ts negativ, either 
aT hOp 8 my, : 

359. Theorem. /f aa> ff, then, if a ts positiv, 
Ge> Pieri a 1S geod, ten. 
~ This follows immediately from § 357 or § 358. 

360. Theorem. /J// either of two integers is even, their 
product is even; of both ar odd, ther product 1s odd ; 
and conversely, if the product of two integers 1s even, 
one of them must be even; uf the product is odd, both 
must be odd. 

Let» a and 6 ber *themstwomuntcrcts: oeectic, 
product. | 

The direct theorem has already been proved in 
§ 305 when £ is positiv. 


a? 


MULTIPLICATION. III 


Suppose fis zero. Then af is zero and the theorem 


is true. §$5320;- 207: 
Suppose # is negativ and equal to — z, wher z is 
even or odd according as f is even or odd. § 288. 
Then af =— (an). 855330, 322, 
But az is even, if either a or z is even, and odd, if 
both @ and z ar odd. § 305. 
Therfor af is even, if either a or f is even, and odd, 
if both a and f ar odd. §§ 288, 182. 


The converse theorem may be proved by the method 
of exhaustion. 

361. Asa product is an integer, we may multiply it 
by another integer or we may combine it with other 
products or integers by plus or minus signs, using 
parentheses to indicate the order in which the various 
operations ar performd. Thus we may hav expres- 
sions like (af) + (70) and (a(? + 7))é. 

362. Theorem. J//f for one or more elements of a 
complex expression containing integers connected by 
signs of multiplication, with or without signs of addition 
or subtraction, or both, equal integers ar substituted, the 
complex expression 1s unchanged. 

- 863, Theorem. af 4+ a7 = a(8 +7). 
For af + ay = (ab,— ab,) + (ac, —ac,). §§ 315,214. 
= (ab, + ac,) — (ab, + ac,) § 259. 
= a(0, + ¢,) —a(6, + ¢,) $§ 310, 239. 
see Oe-bae.O,-1 C.) $531.5. 


= a(8 +7). § 170. 


I12 INTEGERS. 


364. Theorem. ay + fy = (a+ fr. 
365. Theorem. (a+ /)(7 + 0) =a7+ fy + a0 + fo. 


Treating a+ f first as a single symbol, we hav 
(4+ Br +9) = (+ Pr + (4+ 80 § 363. 
= ar + Br + ad + Bo. §§ 364, 214. 

366. Theorem. af — a7 = a(? —7). 


For a8 — ay = af + ay §§ 223, 323, 213. 
= o(2 +7) § 363. 
= a(8 —7) $$ 223, 330. 


367. Theorem. ay — fy = (a — f)r. 

__368. Theorem. (a—/)(y — 0)= ay — fy — ad + feo. 

369. Theorem. v(a+f8+7+---+e+A+p) 

=va+ vB +r +--+. + ve + vA + vm. 
To prove this we writea+ S+ty+---+«e+Atyp 
as a standard sum and apply § 363. 

Thus (a+ A) +) 4° +8) 4942) 
= PB) ery es Pee ANd eee 
=((a+ A) +7) to te) tt yy 

and so on. 

This theorem is the Generalized Left-handed Distri- 

butiv Law for Multiplication and Addition. 

370. Theorem. (a+ P+7+---+e+A+4 py 

=av+ Pytyvt---+ev+t+ dtp. 

Proved from § 364 like § 360. 

This theorem is the Generalized Right-handed Distri- 

butiv Law for Multiplication and Addition. 


MULTIPLICATION, L143 


371. Theorem. (0+ 8+---+4e)\A+p+---+ 09) 
= ah + BA+ eet icA +au+ But reef Kute-- 
+a90+ Bo+---+ xp. 

This is proved by applying §§ 369 and 370 in suc- 
cession. 

This theorem may be stated: The product of two 
complex sums ts equal to the complex sum of all the 
products that can be formd by multiplying an element of 
the first sum by an element of the second. 

372. We may extend the law of § 371 to the prod- 
uct of two algebraic sums. Change all the — signs 
befor integers to +, at the same time changing the 
signs over the corresponding integers. Then by § 371 
the given product may be written as a sum of 
products, the only minus signs being those over in- 
tegers. These may now be removed by §§ 326, 
Bor. 

For example (2—PB—y\(—d+ 6) 

(at P+ y7O+e)=a0 + ae 4 fO+ Pe+yo+72 
= — a0 + ae + BO — e+ 70 —F7e. 

Thus we hav proved the following theorem : 

Theorem. Zhe product of two algebraic sums 1s 
equal to the algebraic sum of all the products that can 
be formd by multiplying an element of the first sum by 
an element of the second, the sign befor any one of these 
products being plus or minus according as the signs be- 


Sor its two elements in the given sums ar lke or unlike. 
& 


114 INTEGERS. 


373. Definition. When a product of two algebraic 
sums is changed to an algebraic sum of products as in 
§ 372, these latter products ar calld partial products 
and the original product is said to be distributed. 

374, Theorem. az = na, 


Forax =a+ta+a-4+..-z times § 301. 
=Ia+i1.a+1.a+.---xztimes 
SSR16, 252. 
=(I+1+1+4.---# times)a § 370. 
U0. $§ 63, 320, 
375. Theorem. af = fa. 
For af = ab, — ad, § 315. 
= ba— ba §§ 374, 2309. 
= (6, —3,)a § 367. 
= Pa. §§ 240, 320. 


This theorem is calld the Commutativ Law for Mul- 
tiplication ('). It may be stated: 

L[n the operation of multiplication the multiplicand 
and multiplier may be interchanged. 

376. Definition. On account of this interchang- 
ability it is proper to giv the multiplicand and multi- 
plier a common name, They ar calld the factors (?) 





(1) For a proof of the commutativ law for multiplication of primary 
numbers see A. M. Legendre, ‘‘ Theorie des Nombres,’’ p. 1 ; Dirichlet- 
Dedekind, ‘‘ Vorlesungen iiber Zahlentheorie,’’ 1879, 4th ed., 1894, 
p. 2; Tannery, p. 58; Beman and Smith, p. 38. 

(2) See Schubert, ‘‘ Encyklopadie,’’ p. 15. 


MULTIPLICATION. I15 


of the product. The first factor is calld the prefactor, 
the second the postfactor (’). 

377. Theorem. very even integer can be written in: 
the form 2a; every odd integer can be written in either 
of the forms 2a + 1 and 2a—1, and conversely, every 
integer that can be written in the form 2a 1s even; 
every integer that can be written in either of the forms 
Bae Pieond 20— 7 25 odd, 

Toa bemirst: place, since Oo = 2.0, the theorem is true 
for the integer zero. 

Second, every positiv even integer can be obtaind 


by adding a number of twos to zero. § 294. 
If the integer equals 0+ 2+2+2+...z times, 
then it equals 27. §§ 209, 301. 


Hence the theorem is true for every positiv even 
integer. 

Next, every negativ even integer can be obtaind 
by subtracting a number of twos from zero. 

Thus it equals 0o—2—2—2—...m times, or 
-— 2m, which equals 2( — 7). 

Hence the theorem is true for every negativ even 
integer. 

Moreover, since every odd integer can be obtaind 
by adding one to the even integer preceding it in the 
natural series or subtracting one from the even integer 
succeeding it, every odd integer can be written in 
either of the forms 2a -+ rand 2a —1. 





(1) See J. W. Gibbs, ‘‘ Vector Analysis,’’ p. 41. 


116 INTEGERS. 


The converse is proved by reversing the steps. 

378. Definition. A complex expression containing 
integers connected by multiplication signs only, with 
parentheses to indicate the order in which the succes- 
siv operations ar performd, is calld a complex product. 
The integers from which it is formd ar calld its ele- 
ments. 


379. Theorem. (a/)z = «4 (7x). 

For (af)x = o8 + a8 + a8 + ---x times § 301. 
=a(P +8+ 8+ ---x times) § 369. 
= a(n). §§ 301, 330. 

380. Theorem. (a/)y = a((7). 

For (afr = (up)e, — (ape, § 315. 


= a(fc,) — a(Be,) $§ 379, 239. 
= a(fc, — Bc,) § 360. 
= 4(97). §§ 315, 330. 
This theorem is calld the Associativ Law for Multi- 


plication (’). 

381. Theorem. A complex product is independent of 
the arrangement of its parentheses and the order of its 
elements. 

The proof of this theorem is formally the same as 
the proof of § 250, the only difference being that 
multiplication signs replace addition signs. 

(1) For a proof of the associativ law for multiplication of primary 


numbers see Legendre, p. 2; Dirichlet-Dedekind, p. 1; Beman and 
Smith, p. 43. 


MULTIPLICATION. 117 


This theorem is the Generalized Associativ and Com- 
mutativ Law for Multiplication ('). 

382. Hence we may speak of a product afyde..-. 
without parentheses and may write the elements of the 
product, a, f, 7, 0, ¢,---, in any order we please with- 
out altering the product. 

383. Definition. The elements of acomplex product 
ar also calld its factors. 

384. Theorem. Zhe product of the left members of 
any number of equalities 1s equal to the product of the 
right members. 

The proof of this theorem is formally the same as 
the proof of § 252. 

385. Theorem. The product of any number of positiv 
integers 1s a positiv integer , the product of any number 
of integers, one of which is zero, ts a zero integer; the 
product of an odd number of negativ integers 1s a nega- 
tiv integer, the product of an even number of negativ 
integers 1s a positiv integer. 

The proof of the first part of the theorem is formally 
the same as that of § 254. 

So also is the proof of the second part, if we write 
the zero integer first. 

To prove the last two parts, we know that the 
product of two negativ integers is positiv. § 320. 

Also, if an integer is multiplied by a negativ integer, 
the sign of the result is opposit to that of the first 
integer. § 320. 





(1) See Legendre, p. 2. 


118 INTEGERS. 


Hence, when we form a product of any number of 
negativ integers by first multiplying two together, then 
their product by a third, this product by a fourth, and 
so on, the products will be alternately positiv and 
negativ, being positiv for two integers. 

The positiv products will then correspond to an 
even number of integers and the negativ products to 
an odd number. 

386. Theorem. // one of a series of integers 1s ZEro, 
their product 1s equal to zero; and conversely, of the 
product of a series of integers is equal to zero, one of 
them must be zero. 

The proof of the direct theorem is formally the same 
as the proof of § 255, if we write the zero integer first. 

The converse is proved by using § 343 repeatedly. 

387. Theorem. (+ a)(f)(+y7)---= + (af -->), 
the plus sign being used in the right member, uf ther ar 
no minus signs or an even number of minus signs in the 
left member, the minus sign being used in the right mem- 
ber, if ther ar an odd number of minus signs in the left 
member. 

This theorem follows from § 326 by a method 
similar to that used to prove the last two parts of 
§ 385. 

388. Theorem. // one of a series of integers is even, 
their product is even y Uf all ar odd, their product ts odd ; 
and conversely, uf the product of a series of integers ts 
even, one of them must be even, of the product is odd, 


all must be odd. %, 


MULTIPLICATION. 119g 


For the proof of the first part of the direct theorem, 
if the series of integers is a, P, 7, 0, ---, the first of the 
products a, (a/)y, ((48)7)0, --- which has an even 
integer for one of its factors is even. § 360. 

The method of § 254 may then be applied to prove 
that the product of the whole series is even. 

The second part of the direct theorem follows im- 
mediately from § 360 by the method of § 254. 

The converse theorem may be proved by the method 
of exhaustion. 

389. Theorem. Zhe product of any number of sums 
2s equal to the sum of all the products that can be formd, 
each having as a factor one and only one element of cach 
of the given sums ('). 

This may be proved by first applying the theorem 
of § 371 to the product of the first two sums, then 
applying the same theorem to the product of this re- 
sult and the next sum, and so on. 

Forexample, (4+f+y7y)0+e)(€+7+4 9) 
= (a0 + Bo+ 70+ ae+ Pe +7el(C +44 9) = ade + 
Bot + yoe + acl + Bel + rel + adn + Bon + yon + aer 
+ Ben + yen + 200 + BOO + 706 + ac + Beb + xed. 

390. By the same method as that employed in 
§ 372 we may extend the preceding theorem to the 
product of any number of algebraic sums. Thus we 
get the following theorem : 

Theorem. The product of any number of algebraic 
sums ts egual to the algebraic sum of all the products 


(1)See G. Chrystal, ‘‘ Text-book of Algebra,’’ 1893, Part I., p. 42. 


I20 INTEGERS. 


that can be formd each having as a factor one and only 
one element of each of the given algebraic sums, the sign 
befor any one of these products being plus, if none of its 
elements or an even number ar immediately preceded by 
minus signs in the given sums, and being minus, if an 
odd number are so preceded (*). 

391. By § 326 the product of a positiv and a negativ 
integer, or of two negativ integers, may be found when 
the product of the corresponding positiv integers is 
known. 

To multiply two positiv integers, or primary num- 
bers, a and 4, we add 4 a’s together. The sum ob- 
taind is the required product. 

Or we may join 4 groups of a objects each so as to 
form a single group. The number of objects in the 
group thus obtaind is the required product. 

392. Multiplication Table(’). In order to make a 
table of the products obtaind by multiplying all pos- 
sible pairs of the numbers 1, 2, 3, ---, 10, except those 
products which may not be represented by the sym- 
bols so far defined, we divide a square into compart- 
ments as in § 74 and write the numbers 1, 2, 3, ---, (0 
along the top and left-hand sides of the square as in 
that article. 

In each compartment we wish to place the number 
which is the product of the two numbers found re- 
spectivly at the left of the row and at the top of the 
column in which the compartment lies. 


(1) See Chrystal, p. 50. (2) See Tannery, p. 68. 


MULTIPLICATION. I21I 


Thus in the first row of compartments we should 
write the numbers that represent the products I x1, 
Meee oe, OL I, 23.3, +. 

In the second row we should write the products 
Boole 2 x2, 2x3, .-- hese ar very easily found as 
follows. We write the natural series of numbers 1, 2, 
3, 4, 5, 6,---. Then starting at I we count to the 
second number. This is 2 x I. Counting to the 
second number beyond this, the number arrived at is 
2x2 (§$ 73). Counting to the second place beyond 
this, we find 2x3, and so on. Thus in the second 
row we write the numbers 2, 4, 6, - 

The numbers to be written in the third row ar the 
multiples of 3, 3X1, 3X2, 3X3, ---, which ar found 
from the natural series by counting by threes. And 
so on. 

It will be noticed that the table thus formd, given 
below, is symmetrical with reference to its principal 
diagonal, which might hav been forseen from the com- 
_ mutativ law for multiplication. 


INTEGERS. 


I22 


MULTIPLICATION TABLE. 



















































































313|O)9|G\¢c 








55 |?/e¢ 
6|6 © 








AL. Wid 





8138/10 





























ChAT TE Rive 


NUMERICAL VALUE. 


393. Definition. ~The numerical value, absolute value, 
or modulus of an integer is, if the integer is positiv 
or zero, the integer itself; if negativ, its opposit. 

We will use the symbol | a | to denote the numerical 
value (*) of @. This symbol is univalent ($§ 161, 153). 

Thus | 5] = 5, |o| =0,|3| = 3. 

394. Theorem. Zhe numerical value of a positiv or 
negativ integer 1s positiv ; the numerical value of a zero 
integer 1S Zero. 

395. Theorem. |a| = | a|. 


If a is positiv, a is negativ. § 164. 
Then |a| =a and|a| =a, S520: 
Therfor | «| = | |. Sis 


iea@iSezero, @. 1S’ Zero. 
tens) ae O1and)| a) =o. 


Therfor | «| Sad |. 
If a is negativ, @ is positiv. 
Then |2| =a and ja] =a. 
Therfor | @ | SG 


396. Theorem. /f two integers ar equal, their nu- 
merical values ar equal. 
(1) The symbol |@| is due to K. Weierstrass. It was used by him 


in 1841. See his works, Vol. I, p. 67. 
123 


124 INTEGERS. 


Let a=f. Then, if a is positiv, f is positiv. § 176. 
Hence: (ia) == a@eand 948. 

Therion) a] = oy 

Similarly the other cases ar proved. 

397. Theorem. /fa=, |a|= | 2]. 

For, by § 396,|4/=|8|. But |2|=|A|. § 395. 
398. Theorem. /f | a. | = 0, 4=0, 

Prove by the method of exhaustion. 

399. Theorem. /fja| =| |, etthera=P ora=§. 
MG haven Veli arex dull answer norel 2 i8) 
Therior a= f. 

simularly, i @e—=0, a0. 

Ther remain four cases to be considered : 

Ist, when a and f ar both positiv ; 

2d, when a is positiv and / negativ ; 

3d, when @ is negativ and f positiv ; 

4th, when a and f are both negativ. 

Caseli, rere a. = ean ys ans 


Hence, since|a|=|8|,a=—. 
Case 2,5 Here a a-and fe 78, 
Hence a= 2: 

Casco icreia 7, 

Therfor a = f. 

Case 4. Herea=f,. 

Therfor a = f. 


In all cases, therfor, either a = f or a =P. 

400. Theorem. Zhe numerical value of the sum of 
two integers 1s not greater than the sum of their numeri- 
cal values. 


NUMERICAL VALUE. 125 


Caser. If a and # ar both positiv, |a| =a and 
1a|=6. 

Then « + f is positiv (§ 206) and |a+ P| =a-+ 9. 

aitetior pate) = la) +18 |. 

Case 2. If either a or f is zero, its numerical value 
is zero and a + f is equal to the other integer. 

Hence {a+ #| and |a| +] | ar both equal to the 
numerical value of the other integer. 

heron) —- P| — | ¢@ || \9 |, 

Case 3. If a and § ar both negativ, |«| =a and 
e\=8 

Then « + f is negativ and |a + 8| = — (4+ f). 

Moreover |a|+|f8|=a+ 8 = —(a+/). 

Therfor |a+ 8|=|a|+]|/|. 

Case 4. If ais positiv and f negativ |a| =a and 
a|=2 

Then a+ P=a—f=|a|—|8|. 

Now, if |a|> ||, |@| —|| is positiv and therfor 
a+ f is positiv. 

Hence |a+f%|=a+f=|a|—| |. 

But |@|—|8]<|a| +18}. $935. 

Therfor |a+8|<|a|+|f|. 

If |a| =| 8|, |@| —| | is zero and therfor a + f is 
zero. 

Hence |a+ 8|=a+P=|a|—|6|. 

Therfor |a+ 8|<[a|+ ||. 

If |a|<|f|, |a|—|| is negativ and therfor a+ 
is negativ. 


Hence |a + 8|= —(#+ #)=|8|—|4|. 


126 INTEGERS. 


But |8|—|a|<|e| +1. 

Therfor |a+ P| <|a|+|P]. 

Similarly we may prove that, if a is negativ and £ 
positiv, |a+8|<|a|+ ||. 

Therfor, in all cases, |a + P|=|a| + |]. 

hates: ja+ 8|p>]a|+] 8. 

401. Theorem. Zhe numerical value of the sum of 
any number of integers is not greater than the sum of 
their numerical values('). 

402. Theorem. Zhe numerical value of the difference 
of two integers 1s not greater than the sum of their 
numerical values. 

Fora—P=a+f. 

Hence |a—8|=|a+8| > [a] +/2]=|a|+18| 

Therfor |a— | > |a|+ | P|. 

403. Theorem. Zhe numerical value of any algebraic 
sum 1s not greater than the sum of the numerical values 
of its elements. 

404, Theorem. Zhe numerical value of the product 
of two integers is the product of their numerical values. 
If a and f ar both positiv, |a| =a and | P| = B. 

Then af is positiv (§ 320) and | af | = af. 

Therfor | ¢8| =|a|| |. * 

If either a or f is zero, its numerical value is zero 
and af is zero. 

Hence | 48| and |a|| | ar both zero. 

Therfor | @@| = | a||#|. 

If a and § are both negativ, |a| = a and | 8| =. 

(1) See Stolz und Gmeiner, p. 64. 





NUMERICAL VALUE. 127 


Then af is positiv and | 48! = af. 

Moreover | a||8| = a8 = af. 

Therfor | a8| =|a||@|. 

If a is positiv and f negativ, ja!=a and | 8| =f. 

Then af is negativ and | a8 | = — (a). 

Therfor | ¢8| =|a|| P|. 

Similarly we may prove the remaining case when a 
is negativ and f positiv. 

405. Theorem. Zhe numerical value of the product 
of any number of integers 1s equal to the product of the 
numerical values of those integers, 

406. Theorem. /f|a|> 1 and +0,|a8|>|/ |. 

Pormuiniah=-tandf--o,|a||ei1 x2), $332. 


Or fae | sie § 404. 
407. Theorem. | aa.| = a, 
For aa is either zero or positiv. § 320. 


408. Theorem. //|a8| < 77, either a or B is numer- 
tcally less than jy. 

For, if neither a nor 8 were numerically less than 
7, we would have | a|=|7| and | @|=|7|. 
Bretices|.a (S| =| 4) [| . $9588 19340, 9350. 
Or | 48 |>r7. §§ 404, 407. 


CHAPTER? ViI° 
DIVISION. 


DIVISIBILITY. FACTORS. QUOTIENT. 


409. Definition. Having given two integers a and 
B, if 8 is not zero and ther is some third integer g such 
that a = gf, ais said to be divisible by /('); if @ is 
not zero and ther is some integer ¢ such that ga = f, 
ais said to be a divisor of f. 

Thus, since 8= 4x 2, Sas divisible byeze. since 
aX! 2" 0, 2 isa civisor oles: 

The reasons for excluding the cases wher § and a 
ar respectivly zero will appear in §§ 422, 424. 

The statement “a is divisible by 8” we will write 
a >>; the statement “a is a divisor of 8” a < £. 

The definitions of the terms “divisible by” and 
‘divisor of’ may, then, be stated in symbols: 

If a = gf, wher f is not zero, then a >> 8; and con- 
versely, if a >> 8, then # is not zero and a= gf. 

If ga = f, wher a is not zero, then 4 < 8; and con- 
versely, if a < , then « is not zero and ga = f. 

The statements a>> 8 and a<f may be calld 
divisibilities. 

(1) See Dirichlet-Dedekind, p. 5. 

128 


DIVISION. 129 


410. It should be noticed that, if @ is zero and 
a=, then « is zero. Similarly, if a is zero and 
Mie 7,09 is Zero. 

Hence by the definitions of § 409 none of the state- 
ments 0 > 0, 0 =< 0, a > 0, 0 < a is true. 

411, Theorem. /f a= of and ¢ + y, then a + xP, 
provided 3 ts not zero. 

For, since g + y and £ is not zero, of + yf. § 338. 

Hence, since a = gf, Tae Vid. Senay 

412. Definition. From the last theorem we see that 
if a = of, wher f is not zero, ther is no integer y, dif- 
ferent from ¢, such that a = 7. 

Hence, if a and # are given, # not being zero, ¢ is 
determind by the equality a = of. 

This passiv number ¢, which depends only on a and 
8 and is such that a = 9, we will call the right-handed 
quotient (") of a and f. 

This quotient we will represent by the symbol 
a:8(*), which is read “a divided by 2.” 

The quotient a: / is, then, univalent (*), when it ex- 
ists, that is, when f is not zero and ther is some third 
integer y, such that a= of. 

The number a may be calld the first element of the 
quotient and / the second element. 

413. Theorem. Jf the right-handed quoticnt a: ex- 
asts and 1s the integer y, then a:8 = @. 

(2) See Schubert, ‘¢ Encykiopadie,’’ p. 16. 

(2) See Schubert, ‘‘ Encyklopadie,’’ p. 16 ; Stolz und Gmeiner, p. 25. 
The symbol : to denote a quotient was used by A. C, Clairaut in 1760. 

(3) See Stolz und Gmeiner, p. 25. 

9 


130 INTEGERS. 


This follows from § 170. 

414, Theorem. /f a= of, wher Fe 7s not zero, then 
a:8=@,; and conversely, ff a:B =, then B ts not 
gero and a= of. 

This follows immediately from the definition of 
right-handed quotient and § 413. 

415. The symbols 0:0, a:0o(') ar both as yet 
meaningless. 

416. Definition. If the restriction in § 412 that P 
should not be zero had been omitted, that article 
would not hav defined the symbol a: f. 

For, if 8 were zero, a would also be zero and a 
would equal yf whatever integer yg might be. 

The symbol a:f, aand # being both zero, would 
then represent any integer. It would not be de- 
fined, made definit. 

A symbol like 0:0, which, tho compounded of two 
definit numbers, would represent more than one num- 
ber, is calld a multivalent symbol (’). 

The symbol 0:0 would in fact represent not only 
more than one integer, but any integer. Such a 
symbol, one representing an unlimited number of 
numbers, is calld an indeterminate symbol. 

Multivalent symbols, if used at all, must be used 
with great care, as the laws which hold for univalent 
symbols do not apply to them. 

417. Theorem. /f ga=f and yo +y, then ya + Bf, 


provided a ts not zero. 


(1) See Schubert, ‘‘ Encyklopadie,’’ p. 17. 
(7) See Schubert, ‘‘ Encyklopadie,’’ p. 17. 


DIVISION. 131 


418, From this theorem we see that the equality 
ga = ? determins the integer ¢, provided a is not zero. 
We will call the integer ¢ the right-handed ratio of £ 
to 4, which we will represent by the symbol (/a, 
which may be read ‘‘ over a.” 

The ratio 8/a is, then, univalent, when it exists, 
that is, when a@ is not zero and ther is some third in- 
teger y, such that ga = f. 

419. Theorem. J/f the right-handed ratio 8/a exists 
and 1s the integer ¢, then p = f/a. 

420. Theorem. Jf ga= 3, wher a is not sero, then 
g = /a; and conversely, if o = 8/4, then « ts not zero 
and pa = 3. 

This follows immediately from the definition of 
right-handed ratio and § 419. 

421. We. will now derive some theorems connecting 
themiceas aepresented by the symbols =, <<, : ;,/, 
which hav been independently defined. 

422. Theorem. /f a>, the right-handed quotient 
a: exists, and conversely. 

The proof of this theorem is formally the same as 
that of § 86. 

The reason is now evident why in § 409 we excluded 
the case of 8 being zero. If we had not done so, we 
should hav 0 > 0, but the right-handed quotient 0:0 
not existing. 

423, From the last theorem it follows that the 
symbol a: as yet has no meaning, unless a > f. 


132 INTEGERS. 


424. Theorem. Jf a < 8, the right-handed ratio 8 ja 
exists, and conversely. 

The reason is now evident why in § 409 we ex- 
cluded the case of «@ being zero. 

425. Theorem. /f asf, then Bxa and 
a/P=a:8; and conversely, if 8B <4, then a> PB and 
Oe 

This is proved in the same way as § 89. 

The proof depends on the commutativ law for 
equality of integers. 

426. We may therfor, whenever we hav either of 
the statements a > f, 8 < a, replace it by the other 
and may replace either of the expressions a:f, a/ by 
the other. 

We will find it convenient in future seldom to use 
the symbol /, but to use in its stead: . 

The symbols >> and < will both be useful, tho we 
will employ chiefly >>. We should always bear in 
mind that the statement a >> # may at pleasure be 
replaced by the statement f < a. 

427, Definition. Having given two integers « and P, 
if 8 is not zero and ther is some third integer g such 
that a = fg, a is said to be a multiple of 8; if a is not 
zero and ther is some integer g such that ag = , a 
is said to be a factor of 9. 

The statement ‘a is a multiple of 8”’ may be written 

a>; the statement “a is a factor of B” a< P. 

The definitions of the terms ‘multiple of” and 

“factor of’’ may, then, be stated in symbols: 


DIVISION. 133 


If a= fy, wher # is not zero, then a>, and 
conversely. 

If ag =f, wher a is not zero, then a<, and 
conversely. 

428. Theorem. /f a= fy and ¢ + y, thena + fy, 
provided f 1s not zero. 

429. Definition. From this theorem we see that the 
equality a= By determins the integer ¢, provided f 
is not zero. 

This activ number ¢, which depends only on f and 
a and is such that 4 = /%y, we will call the left-handed 
quotient (') of f and a. For this quotient we will use 
the symbol f -- a4, which may be read ‘‘f divided into a.” 

The quotient f-- a is, then, univalent, when it ex- 
ists, that is, when f# is not zero and ther is a third 
integer y, such that a = f¢. 

430. Theorem. J// the left-handed quotient B --a ex- 
ists and 1s the integer ~, then B+-a= 9. 

431. Theorem. /f a= fo, wher 8 is not zero, then 
p--a=¢Q, and conversely. 

432. Theorem. /f ap =i and o + y, then ay + 3, 
provided 1s not zero. 

433. Definition. From this theorem we see that 
the equality ag = £ determins the integer ¢, provided 
aisnot zero. We will call this integer the left-handed 
ratio of a to §. For this ratio we will use the symbol 
a\ (?), which may be read ‘ @ under .” 





(') See foot-note p. 36. 
(2) This symbol seems to be due to A. Cayley. 


\ 


134 INTEGERS. 


The ratio a\f is, then, univalent, when it exists, 
that is, when @ is not zero and ther is a third integer 
g, such that ag = f.. 

434. Theorem. J/f the left-handed ratio a\B exists 
and 1s the integer 0, then p = a\f. 

- 485. Theorem. Jf ay =f, wher ais not zero, then 
gy = a\f, and conversely. 

436. Theorem. /f a>, the left-handed quotient 
B --a exists, and conversely. 

437. Theorem. /f a <), the left-handed ratio a\f 
exists, and conversely. 

438. Theorem. Jf arf, then arxfPp and 
B--a=a:8; and conversely, ff a>, then a>B 
and a:B=f a. 

The proof of this theorem depends on the commu- 
tativ law for multiplication. 

439. Theorem. /f ax fpf, thn ax and 
a\8 = Bia; and conversely, ff ax Bf, thn a< Pp 
and B/a= a\f. 

The proof of this theorem also depends on the 
commutativ law for multiplication. 

440. Definition. In virtue of the last two theorems 
we will in future never use the symbols > and <, 
but will use in their stead >+ and < respectivly. 
Similarly instead of #--a@ and /3\a it is customary to 
use a: f and a/. 

In virtue of §§ 425, 438, 439 we will in future in- 
stead of the four terms ‘right-handed quotient,” 
“right-handed ratio,’ ‘left-handed quotient,” and 


DIVISION. 135 


“left-handed ratio”’ use the single term “‘ quotient.” 
Besides the symbols already mentiond for the quo- 


tient of a and # the symbols a + # and 3 ar in com- 


mon use (’). 

441. Definition. The symbols #% and << may be 
used to mean “is not divisible by” and “is not a 
divisor of”’ respectivly. 

442. Theorem. Lvery integer except sero 1s divisible 
by wtself, the quotient being unity. 

In symbols a >> a anda:a=1, ifa+o. 

Prove from §$§ 316, 409, 414. 

443. Theorem. Every integer 1s divisible by unity, the 
guotient being the integer itself. 

In symbols a>} 1 anda: 1= a. 

444, Theorem. Zero zs divisible by every integer ex- 
cept itself, the quotient being zero. 

Inssymbols 0 >> a and o:a—o0, if a+ 0. 

445, Theorem. /fa>> Banda: PB =0, thena=o0. 

Potted) and a: f= 0, d= O- fp =— 0. 

446. Theorem. It follows from the last two theorems 
that the quotient of two integers is zero when, and only 
when, ws first element ts Zero. 

447, Theorem. /f a=, then a>>B anda: P=1, 
provided 2 + O. 

(1) The symbol — for a quotient is found in the ‘‘ Liber Abaci’’ of 
Leonardo of Pisa, This symbol in both the forms %, a/f is probably 





due to the Hindoos. The symbol + was probably originally formd 
as a combination of —and:. 


136 INTEGERS. 


For, ifa=f,a=1-f8. Henceasfanda: P=1. 

448. Theorem. Every even integer is divisible by two. 

For, if 8 is any even integer, 8 = 2a. $377. 

449, Theorem. Vo odd integer is divisible by an 
even integer. 

For, ifia >>, a= 8. 

Hence, if a is odd, g and # must both be odd. 

§ 360. 
450..Theorem. /f az even integer ts divisible by an 
odd integer, the quotient 1s even. 

For, if a> p, a=gPanda:P=g. 

Hence, if a is even, either g or 8 must be even. 

§ 360. 

451. Theorem. /f an odd integer ts divisible by an 
odd integer, the quotient is odd. 

452. The quotient of two integers may be connected 
with other integers by signs for sum, difference, product, 
and quotient, provided that the first element of every 
quotient is divisible by the second element. Thus we 
may hav expressions like 


(4:8) x7): (O28) 
or ((@—f)i7) x O+e)+(0:Q—F+ 9). 


Every such expression is univalent, since every 
sum, difference, product, and quotient in the expres- 
sion is univalent. 

Generally in complex expressions involving quo- 
tients with sums or differences, or both, parentheses 
ar omitted from around quotients. They should 


DIVISION. 137 


never be omitted from around sums or differences, 
when such omission would cause misunderstanding. 

Thus a:8+7:0 is understood to mean 
(a: 8) + (7: 0) and not (a4: (8 +7)): 0. 

453. Theorem. a x § >> Band (a x 8): B =4, pro- 
vided 3 1s not Zero. 

amd =a x: /3. 

nemo x+ 2 > 6 and (a x 8): 2 = dq, 

454. Theorem. A product is divisible by each of tts 
factors, provided that factor ts not zero. 

455. Theorem. /fa>> fi, then (a: 8) x B=a. 

This is proved in the same way as § IIo. 

456. Theorem. /f a>>f, then axy7s>8xy7 and 
(axa) 2 (28 X 7) — 4: B, provided + ts not ecro,; and 
Gums, tf axy > Bx 7, then asP and 
a: 8—= (ax 7): (8 7). 


For the direct theorem, since a > f, 


a= 8 and Cet 
Hence ay = (¢f)7 = g(fy) and (a7): (By) = ¢. 
Therfor ay >> Py erate TEAC (Gor Mas oa ep 


For the converse theorem, since ay >> [7, 
ay = 9(8r) = (Ar and (a7): (Fr) =¢. 

Hence a= oP and Gs = 

theron, «>> 8 and Be (oy) ( 87). 

457. Theorem. /f a>, then yxarxdyx PB and 
x a): (7x P)=4: 8, provided 7x ts not zero; and 
Rorperscly if xX a> yx B, then asf and 
a:B=(7x 4): (7x §). | 


138 INTEGERS. 


458. Theorem. /fa= fPand 8 >7, thenar>s yz and 
Gt 7 asi. 7 ANA CONVELSCLY al a ey a= ye, 
Gi Be Ye — a, 

For the direct theorem, 
since Peevey ieheuiee bite ef 3 Fe 

Hence; since @ "9,40 —.. 07 meand iG te 

Therfor at>y anda:y=f:y7. 

For the converse theorem, let a: y=. 

Thengs —@ 

Henced == anda arc, 

Therion a= 2: 

459. Theorem. /fa>> 8 and B=j7, thena>y and 
O88 == 75) ANA CONUETSELY, Ul 0 > Bee ea 
a:B=a:y7, then B=y, provided ats not zero. 

For the direct theorem, 
since a>>f, a=gPanda:B=g. 

SINCE) Od oe 

Hence CW OPPAN Ge ey 10) 

Therfor a> Fand ae B= an, 

For the converse theorem, let a: f= g@. 

Thentars ig 

Hence a = ¢f and a= gy. 

Hence of = gy. 


Moreover, since @ is not zero, a: f, or g, is not zero. 

Therion =". 

460. Theorem. /fars BP and 8 >y7, thena sy and 
a:7=(a:8) x (Bz. 

Since te Pag — Op ees eee 

(1) See Tannery, p. 98; Dirichlet-Dedekind, p. 5. 





DIVISION. 139 


Sitewen it — yp. and Pp: 7 ='y. 

Hence = g = ¢ (77). 

Hence ay ea UUG a= OY. 

Therfor Li Biel G28 LSS Tie eae 


461. Theorem. /f we hav a series of divisibilities of — 
the same kind, as a>> B >> 7 D> ODS eS, any integer 
on the serves 1s divisible by any following integer. 

The proof of this theorem is identical in form with 
that of § 196. 

462. Theorem. /f we hav a series of equalities or 
diwvisibilities of the formas fey yds eS.---, any 
integer in the series 1s atvisible by any following 
integer. 

a5seecheorem, If (a-> 8 ~and \B > 7, ~ then 
Org 0 ad (o:7) (pin) —a2p; and conversely, 
ee 7, Od Os = 7 en a o> PB and 
gene) (9 +7). 

For the direct theorem, since a > P >7, 

Meee eand.¢ 57— (a: 8) x (8:7). 

ilemoria=4 > p-7 and:(a@-7):(8:7)— a: p. 

For the converse theorem, 
since aro a= oy and a:7 = ¢. 

pile fe yy and 77. y. 

Ceean ya > f: 7) 0 > Y. 

Hence Cee i. 

Or Th easel 

Therfor, by the direct theorem, since 4>> P>>7, 

Cie raag Oy i018 3) 


140 INTEGERS. 


464. Theorem. /f a> and Boyz, then 
a:y > a:p and (az7y):(aif)= B27, provided ats rol 
zero, and conversely, if a>>7, a>> 8, and a:7 > a:8, 
Thess ty ARO Sy Ne ga) (rs) 

465. Theorem. /f a>>f and y=0, then ay > fo 
and (ay): (80) = a: 8, provided y+ ts not zero; and con- 
versely, if ay>>BO0 and y=0, then ax>P and 
a: 8 = (ap) : (0) 

For the direct theorem, 

SINCE tees s ape gee yay eee 

Since =O; es fae—r0, 

Therfor ay >> 80 and (ar): (87) = (ar): (80). 

Also (ay): (80) = a: B. 

For the converse theorem, 


since 70, Sy = po. 


Hence, 

since ay 3 9, ay > By and (ay): (87) = (a7) :(0) 
Therfor a>>fP and fo8 eye 
Also ar B= (apt (Boy, 


466. Theorem. /f a> and 7 >> 0, then ay > Bo 
and (ay) : (90) = (a: B)x (7:0). 
If y= 0, ay = O and az > Po. 
Also 730 = 0 and (ay) : (80) = o. 
Therfor.(ayya.(0) == 4 anew eng): 
If 7 + 0,.since a > , ay >> Br and (ay):(8y)) =a: f. 
Since yas 0s Br o> Po and (87): (Rd) => 210. 
Therfor ay >> Bo 
and (07):(30) = [(ar):(2)] x [(6r):(08)] 
=f SNS XA yore §§ 460, 331. 


+f 


DIVISION. I4I 


467, Theorem. /f a>>f, then ay P and 
(ay): 8 = (4: 8) x 7. 

This follows immediately from § 466 by supposing 
Oe at: 

468. Theorem. /f a>, then yarrB and 
(7a): B=7x (a: 8) 

469. Theorem. /f a=, y=0, and B>y7, then 
eee Ode. Ora 8 2 0 and conversely, 17 
a>y7, P>0,a:7=8:0, andy=0, thna=8; if 
ep Oa i 2120, and a= 1p, then += 0, 
provided ts not zero. 

For the direct theorem, 

PnicemeC p> 7, @ >> 7 and\a fy== Ps 7. 

SYNGE 7) S27 SSUh pe Serb oie — wore 

r=fi0 


Therfor aArr= 
For the first converse theorem, 

since B>0=7, CEE ee Ee 
encom Come S 7 0m) | Oye 0 ry 
Therfor a= fp. 


Similarly the remaining theorem may be proved. 
470. Theorem. /f a>, y=0, and Boy, then 
Meee 0, Os Ftp 20, aNd (a: 7) 3 (8.70) eia sy 8 
and conversely, if a>> 7, B > 0,a:7 >> B30, and 7 =O, 
then a >> fp. 
For the direct theorem, 
since ey ee a, 
and (dary) (Psy) =a 282. § 463 
Bitcomet 0.0 o> Osand p27 =—=99": 0. 
Therfor ss ye ead PR 


I42 INTEGERS. 


and (air): (Bin)=(@i7) 2 (820). = $450. 
Also (Case Ges 0 y= na, 
For the converse theorem, 

SinCom e107) inp ravi ye) sp 
Piencewsince ce enor OF BS eRe 
Therfor ie oh 


471. Theorem. /f a=, 7 >>0, and Bo>>yx, then 
Oo 7 EO, ese 0, ea etn ee ey ee 
provided f ts not zero, and conversely, if a>> 7, 8 > 0, 
a:7<B:0, anda=f, then; > 0, provided P ts not 
Zero. 

For the direct theorem, 
since a=P>y7, a>>7, anda:7y=f :7. 

SINCE tp > GeO 10, Oe ee, 


and COSORERIG EE Reape 0. 
Therfor Cs ee GeO. 
and (38: O)eni Choa) ee ee 
Also (G5r20) ec het) canner: 
For the converse theorem, 
since CREE eS ree Ie ye ein oe 
Pence since 2 10 eo ean 
Therfor ye IOs 
472. Theorem. /f de Dish ‘fee Oe and 


B71, then ies10: O30 eee ana 
(a: 8): (B27) = (a: 8) x (7:9). 

Since yp 70 ee ee Oe ee 

(8:8) 3{8 37) =7 20. 

Since Lip ily Ramen VES aa) Shans: 

(0020) Fa(3 x0) = es, 


DIVISION. 143 


Therfor DAN So Tepe rial: 
(2:0): (8:7) = [(@: 9): (@:9)] x (8:9): @:7)] 
= (a: f) x (729), 


473. Theorem. Jf a>> 8, 7 0, and a: B=7: 4, 
then a0 = Br; and conversely, if a> PB and ad = fy, 
theny >O anda: B=y7:0, provided 0 is not zero. 

For the direct theorem, 
since a>, a0 >> Pd and(ad): (0d) =a: f. 

pice yae> 10127. "p0m and (by) 2(f0) — 7 +0. 

ecm INCOR: 1 9) 701-10) 20 Jes (90) 197), 21( (00). 

Therfor eye 

For the converse theorem, since a > f, 

ad >> Bd and (ad) : (80) =a: f. 

Hence, since ad = fy, 

ihe Se ig and (7) : (80) = (40) : (89). 

Hence 70 aTiCay geo —"( 7 ye 30). 

Therfor Cea =F erO8 

474. Theorem.. /f ar>> 8, 7 >> 0, and a: B77: 0, 
Pie tou py and) (do) (Bry — (a 28) 230) 5 and 
conversely, if y>> 0 and ad> Br, then ar>BP and 
e720: 

For the direct theorem, 
sincema >), 40 > 90. and (a0) <.(f0)— a: £. 

pilceyy > 0, py >> po and! (7): (fd) = 7: 6. 

Pence, since : 9 2772/0, (ao): (80) > (87), (80) 
and [(ad) : (38)] :[(8r) : (8)] = (a: 8) :(7 +8). § 460. 

Therfor ad >> Py and (ad) : (Br) = (a: f) : (7: 9). 


For the converse theorem, 


since es Om tye polalt{py)i (20) = 7.2.0. 


144 INTEGERS, 


Hence, since ad > fy, 
ad >> Bdand (ad) : (80) > (Gy) : (Be). 

Therfor esse and 9 == (ape (een 

Therfor Os Beay ne 

475. By comparing §§ 409-474 with §§ 75-128, 
the analogy may be seen that exists between the 
theory of products and quotients on the one hand and 
that of sums and differences on the other. Also 
certain points of divergence between the two theories 
will appear. 

476. Theorem. /farsf, anda: PB >7, thena> fr 
GUO LOE (i) a aa lata 

This follows from § 474 by supposing 0 = 1. 

477. Theorem. /far>> 2 andy >a: 8, then y Poa 
GRGNTS Vic a7" Cea) ee 

478. Theorem. /farsy and B77, thna+tProrx 
and (a+fP):r=(a:r)+(Pir)> and conversely, 
ff Ge and: TE ee het a ee, 

(2+ 8): 7=(4:7) + (8:7). 

For the direct theorem, 
SINCE a> 7.10 —— Oa soe ec, 

SUNCE Gp a yan 3 een. 

Hence a+f=o7+yr=(9 +47. 

Therfor a+ 8 >yand(a+ P):7,=o+y. 


Or (2+ P):r=(@:7)+ (8:7). 
For the converse theorem, 
since Looe, Cer, a 


Sincea+Pro7,a+ 8 = dy. 
Hence (a+ ~)-—a= dy — gy =(¢ — pyr. 


DIVISION. 145 


Or B= (¢ — yr. 

Therfor 2 >> 7 and by the direct theorem 

eS Vict (Obagi at), 

This theorem is the Distributiv Law for Division and 
Addition. 

479. Theorem. /farsy and B>7, thena—Bro7 
and (a—f):y=(a:7)—(8:7)5 and conversely, 
yee, did a—P>y, then poy and 
(a— 2) :7= (4:7) —(8 is FB randa—B yz, 
then a>>yand(a—f):7=(a:y7)—(8:7). © 

t5UmeLueoreM.. Jf aS oS cand ik 
and ar any two integers, then ha+ pBr>>y and 
(4a + mB) 27 = A427) + HB 27) (?). 

Bolemcheorou. yy a Bey and A 
and wt ar any two integers, then ha—pBr>y and 
(1a — 8) 27 =2 (027) — w(3 +7). 

482. Theorem. /f ar and yr>d,_ then 
ad + Pr >> Bd and (a0 + Br) : (80) = (a: 8)+ (7:2). 

Bincog eae Pp, 20 > po.and (a0)': (80) — « =p: 

A eom = One 10, and (07) (00) ==-7 «0. 

Therfor ad-+ fy > Bo 
and (ad + Pr) : (B0) = (ad) : (88) + (87) : (92). 

Also (ad + Pr) : (80) = (a: f)+ (7: 0): 

483. Theorem. /f a>>f8 and yr, then 

ad — fr >> 20 and (ad — By) : (80) = (a: B)—(7: OD. 


484. Theorem. /f a> 8 and a and f ar both positiv 
or both negativ, then a: 3 is positiv,; of either a or 3 is 





(1) See Legendre, p. 3 ; Chrystal, p. 39. 
10 


146 INTEGERS. 


positiv and the other negativ, then a: ts negatv ; 
and conversely, of the quotient a: 3 ts positiv, aand 8 ar 
either both positiv or both negativ; if a: Pts negativ, 
either a or 3 is positiv and the other negativ. 

Suppose a >> # and that @ and f ar both positiv. 

Then a= op anda: 8 — 

Now ¢g is either positiv, zero, or negativ. 

But ¢g cannot be zero, since then $F; and hence a, 
would be zero. 

Similarly g cannot be negativ. 

Therfor ¢ is positiv. 

The other parts of the direct theorem ar proved in 
a similar manner. 

The converse theorem is proved by the method of 
exhaustion. 
485. Theorem. Jf ac f, then —arrf and 
(—a) :P=—(a:f). 

INCE Cts ed = ACen a 

Hence —a=—(y8)=(—9)8 

Therfor —a>> Band (—4):8=—¢g=— (a: f). 

486. Since the expressions (— a) : 8 and — (a: #)ar 
equal, the parentheses may be omitted and both be 
written — a: f. 

487. Theorem. Jf a>>f, then ar>>—f and 
a:(—p) = —(a:f). 

488. Theorem. /f af, then —a>—f8 and 
(— a) :(—A) =a: 8 

489, Theorem. a >> — 1 anda:(—1)=— 


DIVISION. 147 


490. From the theorems of §§ 170, 485, 487, 488 

we derive the following equalities : 
(+0):(4+8)= +478, 
(—a):(4+ #)=—4:8, 
Caan(— 6) = 24:8, 
(—a):(—8)=+4: f, provided a> P. 

These equalities constitute the Rule of Signs for 
Division, which is usually briefly stated: /xz division 
like signs giv +, unlike sigus — . | 

In particular we hav (— a) :d =a: (— 6) = —(a: 0) 

and (— a) :(— 6) =a: 4, provided a > 4. - 
"491, Theorem. /f a>fP and jy ts positiv, then 
a:7>P:7, provided a and f ar both divisible by y. 

ele ree ya alld. 2) 

SiiGesoee te gi Ya. dild qian, 

Since 1,07 => yy. 

Therfor, since 7 is positiv, g > y. S345. 

Or Ls Da aes oag gi 

492. Theorem. /f a>f and jy ts negativ, then 
—a:7<B :7, provided a and 8 ar both divisible by yx. 

493. Theorem. /f a>, and B and 7 ar positiv, 
then y:a<7:f, provided x ts divisible by both a and 3. 

Sineee) > 2, 7) — Coeand, 7: a == ¢, 

Bilge ee — Ypeand 7p — 7. 

Hence ga= 78. (1) 

Since f is positiv, a is positiv. § 201. 

Since # and 7 ar positiv, y is positiv. 

Hence, from (1), since a > # and aand y ar positiv, 


O<yX. § 353. 


148 INTEGERS. 


‘Cherloriy maa <2. 7 34 
494. Theorem. /f a>f and y=0, then, tf 7 ts 
positty, O39 > BOS Ur ue aND, Gag 80: 
This follows from §§ 491, 459, 492. 
-495. Theorem. /f a=f, y> 0, and a and 0 ar 
positiv, thna:7<B:0. | 
496. Theorem. Jf arf, }a| 18 and 
|| :|B]=[2: BI 
Since dP, a= en anu as 
Hence |a|=|¢2|/=|¢|| 2]. 
Therfor |@|>+|8| and |a|:|/P|=|¢|=|a: B. 
497. Theorem. /f ax integer ts not zero, cach of tts 
factors, except the integer itself and its opposit, ts 
numerically less than the integer. 
Let a> 8, whereas and | miles) Bn 
Then a = gf and |a|=|¢8|=|¢||8|. 
Now either |¢| =0,|g|=1, or |¢|> 1. 
If |¢|=0, p= 06 and a= of =o. § 308. 
But @=b OF iTencesip) aaa 
If|g|=1, |a|=|P]. 
Hence in this case f is either 2 or its opposit. § 399. 
Ther remains only the case when | ¢ | > 1. 
In this case | 92 | > |? |. § 400. 
enathic: pal S184, or | P| <a]. 
Hence the theorem is proved. 
498. Theorem. Vo integer except zero is divisible by 
an integer numerically greater than ttself. 
499. For example, the symbols (— 3:49) 
(— 4) :(— 8), 5 :(— 7) ar, as yet, meaningless. 


DIVISION. I49 


500. Theorem. Uvity has no positiv factor except 
unity. 

501. Theorem. /f »= ap and 1 <|a|<|v|, chen 
1<|P1<|>| 

For, since ya kT |v | > 0. (1) 

Now |»|=|43/=[4| [8]. (2 

If | 8 | =0, |»| =o, which contradicts (r). 

If |8|=1,|»| =|], which contradicts the hypoth- 
esis. 


ee pe ||) |} sla} | and “hence |'a|= 1, 
which contradicts the hypothesis. 
If |8|>|»|, then, since | af| > | 8! ($ 406). 


| @2| > |» |, which contradicts (2). 
Mherfor v8 |< |v |. 
502. Theorem. /fa>> 8 and 8 >a, |a|=| |. 
By the hypothesis neither « nor is zero. 


Pieticewsmce a* > (Bla = fal, § 497. 
Peso aeinee 4s a | al =| Bp |. 
Therfor fal = 12: - 


503. Theorem. /f a and f ar any two integers, ex- 
cept that 2 may not be zero, an infinit number of pairs 
of integers h, can be found such that iB <a< 13. 

Consider the multiples of / 

) = 3B, ee 2)8, eS 1), Op, 1f, 2B, 32, i a 

These multiples increase from left to right, if # is 
positiv, and from right to left, if # is negativ. Among 
them ar found the products af and (— a). 

Ther ar three cases to be considerd. 

Caser. a=o. Then —|8|<a<|f|. 


150 INTEGERS. 


This givs, if 8 is positiv, (—1)P<a<1x B; 
if B is negativ, 1 x B<a<(—1)f. 
Case2, |B|=1. Sinceea—I1 ca<a+tl, 
we hav (a—1)|8|<a<(4+4+1)|/|. 
This givs, if fis positiv, (4—1)B <a<(a+ 1I)f; 
if 8 is negativ, (— (4— 1) @<a<(— (4+ 1))f. 
Case 3 GO Dang ee Let e 
o<|4|<]@||8|(). 
This case may be divided into four subcases. 
1°. a positiv, f positiv. The above result becomes 
Ore 0 BTL en 4d a 
2°. a positiv, PB negativ. Here 0-S<a<(—a)f. 
3°. anegativ, § positiv. Here o<—a<(— a), 
which givs o> a> af, or aB<ca<o-f. 
4°) Oo Neca eo Me cavemee Tere 
(—aB<a<o-fp. 
In each case, then, we hav found two multiples of 


8 between which a lies. The coefficients of these 
multiples ar a pair of numbers /, yz, such that 


IB<a< pp. | 
Again, if 8 is positiv and « is any integer less than 
A and » any integer greater than pp, «8 < AP and 
Hoa Yb, Bilence,Kks <a ue, 
Similarly, if P is negativ. 
Therfor an infinit number of such pairs of integers 
can be found. $§ 2324537: 





(1) See Stolz und Gmeiner, p. 25. 


DIVISION. 151 


504. When a and f ar positiv, is positiv. The 
preceding theorem then givs as a particular result the 
following : 

Theorem. /f a and b ar two primary numbers and 
a>b, a primary number n can be found such that 
Miata 

This theorem is sometimes calld the Axiom.of Archi- 
medes ('). 

505. The theorem of § 503 furnishes us a means for 
determining whether a> ? and, if a> f, for finding 
the quotient a : f. 

If ais a multiple of /, it must be one of the mul- 
tiples that lie between two particular multiples 2? and 
8, which ar easily found. We hav then to consider 
only the limited number of multiples that lie between 
AB and wP. If one of these is equal to a, a >> P and 
a: is the coefficient of that multiple. If none of 
these is equal to a, a } f. 

Thus, to find whether 7 is divisible by 2, we need 
consider only the products that lie between 0 x 2 and 
pee Ge AVE Ble Xs = 1252 X 2 4, 30K 2 = 6; 
eeceson Nowell 7-74, 2x 2 > 4 « 2. Hence 
all higher products ar greater than 8, and therfor 
greater than 7. 

Since, then, ther is no multiple of 2 which equals 7, 


he 2: 


(1) See D. Hilbert, ‘‘Grundlagen der Geometrie,’’ 1899, English 
translation, ‘‘ The Foundations of Geometry,’’ 1902, by E. J. Town- 
send, p. 25. 


152 INTEGERS. 


The symbol 7:2 is, then, at present meaningless. 
Similarly the symbols 7:(— 5), 4:3, (— 5):(— 4) ar 
meaningless. , 

506. Definition. Division. The process described 
in §505 for finding whether a is divisible by /, and, if 
divisible, for finding the quotient, is calld division. 

Thus the quotient a:/, when it exists, is obtaind 
from a and by an operation, the operation of divid- 
ing a by f. In this operation a is the operand, the 
passiv element, or dividend, and / the operator, the 
activ element, or divisor. 

When a@ and far the primary numbers a and 4, 
their quotient a: 0, when it exists, may be obtaind 
by dividing a group of @ objects into 4 groups, the 
numbers of objects in the latter groups being the 
same, we v. 

Or we may divide a group of @ objects into groups 
of & each, the number of these groups being @: 0('). 

These results follow directly from § 455. 

507. By § 490 the quotient of a positiv and a neg- 
ativ integer, or of a negativ and a positiv integer, or 
of two negativ integers, may be found when the quo- 
tient of the corresponding positiv integers is known, if 
the latter quotient exists. 

508. Division Table. In order to make a table of 
the existing quotients of all possible pairs of the num- 
bers I, 2, 3, ---, 0, we divide a square into compart- 
ments as in § 74 and write the numbers I, 2, 3,---, 





(1) See Tannery, p. 17. 


DIVISION. 153 


along the top and left-hand sides of the square as in 
that article. 

In each compartment we wish to place the number 
which is the quotient of the two numbers found re- 
spectivly at the left of the row and at the top of the 
column in which the compartment lies, if this quotient 
exists. | 

Thus in the eighth row of compartments we should 
write the numbers that represent the quotients 8:1, 
beta te Ou 4, + 

To find these quotients we may use the multiplica- 
tion table to advantage. We notice that in the com- 
partments of that table the 8’s lie in the first, second, 
fourth, and eighth columns. The 8 of the first col- 
Miieioeinetie ciehth tow - hence $:1— &.~ The 8 
of the second column is in the fourth row; hence 
8:2=4. Ther is no 8 in the third column; hence 
ther is no quotient 8:3. The 8 of the fourth column 
is in the second row; hence 8:4= 2. And so on. 

Hence the eighth row of our new table will contain 
Pi Mmumibers+o, 4,2, lin the first, second, fourth, and 
eighth columns, and none in the others. 

In a similar manner we may form the rest of the 
table, given on the next page. 

509. Theorem. TZhe operation of division 1s inverse 
to multiplication ('). 

This follows immediately from § 453. 





(1) See Schubert, ‘¢ Encyklopadie,’’ p. 16. 


154 INTEGERS. 


DIVISION TABLE. 





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510. Theorem. J/ultiplication is inverse to division. 

511. Definition. The operation of multiplication, 
which was defined (§ 297) as a repeated performance 
of the direct operation addition is also calld a direct, 
or synthetic, operation. 

On the other hand, in division, having given two 
integers, of which the second is a factor of the first, 
the object is to find a third integer which multiplied 
by the second makes the first. Every question in 


DIVISION. 155 


division is, then, turnd into a question of multiplica- 
tion. For this reason division is calld an indirect, or 
analytic, operation. 

For reasons the same as above given a x f# and 
a: ar calld respectivly synthetic and analytic combi- 
nations of the integers a and f. 

The opposition in nature between multiplication and 
division may be shown by the equality 7 = af. 

If a and f ar given to find 7, we get the result by 
multiplication. 

If 7 and f ar given to find a, or 7 and a given to 
find 8, we get the result by division, the operations in 
these two cases being of the same nature because of 
the commutativ law for multiplication. 

To distinguish multiplication and division on the 
one hand from addition and subtraction on the other, 
the former ar calld the operations of the second 
step ('), or second degree, the latter the operations of 

the first step, or first degree. 
= The operation of multiplication is always possi- 
ble. The operation of division is possible only when 
the passiv number is a multiple of the activ number. 


512. Theorem. When the operation of division shows 
that a} 8, a patr of integers y, 4 can be found, and 
only one pair, such that gf ts the greatest of all the 
multiples of which ar less (?) than a, xB the least of 


(1) See Schubert, ‘‘“Encyklopadie,’’ p. 14 ; Stolz und Gmeiner, p. 9. 
(?) See Chrystal, p. 42. 


156 INTEGERS. 


all the multiples of 8 which ar greater than a, and 
Y¥-g=+17('), according as f 1s positiv or negativ. 

We hav AB <a < pf. § 503. 

Of course all those multiples of which ar less than 
43 ar also less than @ and those greater than pf ar 
greater than a. 

Now of all the multiples which ar less than a, from 
A3 up, pick out the greatest and let its coefficient be 
gy. Then gf is the greatest of all the multiples of f 
that ar less than a. 

Also, of all the multiples which ar greater than a, 
from “8 down, pick out the smallest and let its co- 
efficient be y. Then ¥f is the smallest of all the 
multiples of # that ar greater than a. 

We hav, then, 98 <a <7, and hence of < 7. 

Now gf and 7 ar consecutiv multiples of 7. 

For, suppose ther existed an intermediate multiple, 
YB, so that 98 < $3 <yB. 

Then either $2 <a or $8 >a. 

If ¢8 <a, then vf would not be the greatest of all 
the multiples of 8 which ar less than a. 

If Jf > a, then yf would not be the smallest of all 
the multiples of # which are greater than a. 

Hence ther is no multiple of # between of and yf 
and these two multiples ar consecutiv. 

Therfor, if 8 is positiv, y= ¢ + 1andy—g=1; 
if 2 is negativ, Y¥=o—Ii1andy—¢g=-—I. 

Only one such pair of integers can be found, since 





(1) See Stolz und Gmeiner, p. 24. 


DIVISION. 157 


ther is only one greatest, and one least, among a set 
of integers, no two of which ar equal. 

513. Definition. Besides the process previously calld 
division (§ 506), the process for finding the integers ¢ 
and y, when a } #, as described in the last article, is 
also calld division; «@ is the dividend, / the divisor, 
gy the lower quotient, y the upper quotient. 

To distinguish division in the sense of § 506 from 
division in the present sense, the former may be calld 
exact division and the latter approximate division ('). 
In exact division the dividend is said to be exactly 
divisible by the divisor. 

514. Definition. When a is divided by f giving a 
quotient g, either exact quotient, lower approximate 
quotient, or upper approximate quotient, the difference 
a — of is calld the remainder (’). 

515. Theorem. When «a is divided by B giving a 
guotient ¢, either exact, lower approximate, or upper ap- 
proaunate, if p is the corresponding remainder, then 
a= 98 + p. 

516. Theorem. Jz exact division the remainder ts 
zero, in approximate division the remainder corresponda- 
ing to the lower quotient is positiv, the remainder corre- 
sponding to the upper quotient ts negativ. 

For in the first case a = ¢f, in the second a > of, 
in the third a < ¢f. 

517. When in approximate division the words 


(1) See Stolz und Gmeiner, p. 25. 
(2) See Chrystal, p. 42; Tannery, p. 80; Stolz und Gmeiner, p. 25. 


158 INTEGERS. 


) 


“ quotient” and “remainder” ar used without quali. 
fication, the lower quotient and positiv remainder ar 


usually meant. 


518. Theorem. When a is approximately divided by 
B, the sum of the numerical values of the positiv and 
negativ remainders ts equal to the numerical value of f3. 

Let p and a be the positiv and negativ remainders 
respectivly. 

Then |p|+|¢|=ep+¢=p—<a. 

Buta=o8+panda=yf+ a. 

Hence gB+tp=y7P +e. 

Therfor Po Ne Oe awe 
= + 1-f, according as f is positiv or negativ, 

= + f. 

Hence |p| + || =[BI. 

519, Theorem. When ats approximately divided by 
8, both remainders ar numerically less than i. 

This follows from §§ 518, 264, 234. 


520. Theorem. /f, when two integers ar approxi 
mately divided by the same integer, the positiv remain- 
ders ar equal, then the negativ remainders ar also equal, 
and conversely. 


521. Theorem. /f a=of+p, wher P+ 0 and 
|e|<|B\|> then, of p ts zero, y ts the exact quotient ot 
a divided by 8; if p is positiv, y ts the lower approxi 
mate quotient, and, if p is negativ, y ts the upper ap- 
proaimate quotient, in each case p 1s the corresponding 
remainder. 


DIVISION. 159 


First, ifo = 0, a= 98 + p = of. 


Therfor Cag, 
Second, suppose g is positiv. 
Then gB <a. | 


Moreover 9 is the greatest of all the multiples of 
8 which ar less than a. 

For (gp +1)8> gf, the upper or the lower sign 
being used according as 9 is positiv or negativ. 

S$ 332, 336. 

Also, since +P>p, pPPEP> GB + p. 

Or (Cat ie oO 

Hence the next higher multiple of § after gf is 
greater than a. 

Therfor gf is the greatest of all the multiples of f 
which ar less than a, and ¢ is the lower approximate 
quotient of a divided by P. 

Similarly, if o is negativ, it may be shown that ¢ 
is the upper approximate quotient. 

In each case we have 4 — of = p and therfor p is 
the corresponding remainder. 

522. Theorem. /faand f ar any two integers, except 
that 2 may not be zero, then acan be written in the 
form o8 + p, wher |p|<|f|, 2 one, and only one, way, 
when ats exactly divisible by 3, and in two ways, and 
no more, when ais not exactly awisible by 8, one with p 
positiv and one with p negativ. 

It follows from § 515 that one such way exists 
when a is exactly divisible by £, yg being the quotient 
a:f8 and p being zero, and two ways when a is not 


160 INTEGERS. 


exactly divisible by 8, in one way ¢ being the lower 
quotient and p the corresponding positiv remainder, 
in the other ¢ being the upper quotient and p the cor- 
responding negativ remainder. 

To prove that no other such form of expressing a 
can be obtaind when «@ is exactly divisible by P, 
suppose $f + p is any such form. 

We hav also a= of. 

Hence ¢f + p = of. 

Therfor p = 98 — $8 = (yg — $) 8. 

Hence: o> /pae But) 6) 23/6) el heron Gc: 

§ 498. 

Hence ¢# is the exact quotient of a divided by f. 

S.52i: 

This form is, then, the same as the other. 

Next suppose a is not exactly divisible by 9 and 
let v8 + p be any form of expressing 4. 

Then p= 0. “For,at pi==0,'a would besexactly, 
divisible by f. 

Hence p is either positiv or negativ. 

If p is positiv, g is the lower approximate quotient 
and p the corresponding remainder. § 521. 

If p is negativ, ¢ is the upper approximate quotient 
and p the corresponding remainder. 

Our theorem is, then, proved. 

523. Theorem. /f « zs exprest in the form of + p, 
wher |p| <| |, then @ ts or ts not exactly divisible by 
8, according as p ts or 1s not Zero. 

524, Theorem. Jf « be approximately divided by f, 


DIVISION. IOI 


giving a quotient g and remainder p, then, when uf ts 
divided by 20, a quotient, lower or upper, will be ¢(') 
and the corresponding remainder pb. 

For we hav a= of + p, wher | o| <| |. 

Hence af = (gf)0 + 8. 
Or af = o( fA) +8. 

Also, since | 6| is positiv, |||@|<|P||@|- 

That is, | eA | <| A). 

Therfor, when af is divided by £6, a quotient, 
lower or upper, must be ¢g and the corresponding re- 
mainder 0. S352 1 

525. The most important particular case of the pre- 
ceding investigation is when @ and f ar both positiv 
integers, or primary numbers. 

Suppose @ and J ar any two primary numbers and 
qe U. 

If a is exactly divisible by 4, a= q0. 

If a is not exactly divisible by 4, let g be the lower 
quotient, 7 the upper, 7 the positiv remainder, and — s 
the negativ remainder. 

Then a=gb+7,a=tb —s,r+s=46, and rand 
s ar both less than 4. 

We hav also the following additional theorems for 
primary numbers, or positiv integers. 

526. Theorem. /f a and f ar positiv, a>, and 
ats B, then both approximate quotients obtaind by a- 
viding a by B ar positiv. 


(7) See Stolz und Gmeiner, p. 34. 


Pert 


162 INTEGERS. 


Let ¢ be the lower approximate quotient and p the 
corresponding remainder. 

Then a4 = 98 + p, wher a> B> p. 

Hence gf = a — p, wher a> p. 

Hence 9g = (a — p):f.. 

Since a> p, @ — p is positiv. 

Since, therfor, g is the quotient of two positiv in- 
tegers, @ 1s positiv. 

Similarly we may prove that the upper quotient is 
positiv. 

527. Theorem. Jf a and £ ar positiv, a> f, and 
ats 8, and the lower approximate quotient ~ obtaind 
when a is divided by 8 ts greater than the corresponding 
remainder, then, when ats divided by ¢, the lower ap- 
proximate quotient 1s 8 and the remainder the same as 
when ats divided by fi. 

Let p be the positiv remainder obtaind when a is 
divided by /. 

Then a = gf + p, wher p< Q. 

Hence a= fe + p. 

The theorem then follows from § 521. 

528. Theorem. Jf a and f ar positiv, a>, and 
ats 8, and the lower approximate quotient ¢ obtaind 
when ais divided by 2 ts greater than P, then, when a 
is divided by o, the lower approximate quotient is B and 
the remainder the same as when ats divided by £. 

Porro 3 anday Se 

529. Theorem. Jf a and f ar positiv, a> B, and 
at 2, and the upper approaimate quotient y obtaind 


DIVISION. 163 


when ais divided by 8 ts greater than the corresponding 
remainder, then, when ais adiwided by y, the upper ap- 
proaimate quotient 1s B and the remainder the same as 
when ats divided bv BB. 

530. Theorem. Jf a and $8 ar positiv, a> PB, and 
ats 8, and the upper approximate quotient y obtaind 
when ats divided by B ts greater than f, then, when «a. 
is divided by ¥, the upper approximate quotient is 2 and 
the remainder the same as when «1s divided by f. 

531. Theorem. /f/1 << $ <aand ¢ ts the exact or 
lower approximate quotient obtaind by dividing « by f, 
then both » and y +1 ar less than a and (g +1) >a. 

Since 9 => 1, hes © 


Hence gf=¢ xX 2. §§ 330, 335. 
Or ~YP>o t+ ¢. 

Now aS of. 

tea, ioe Hie § 501. 
Hence g+o>o+l. eZ LO! 


Hence in this case we hav 
a=¢ofPSo+e>¢t+i. 


Therfor a>got+l. 


Moreover, since g++ 1>9, (9 +1)8B >¢P. § 332. 
Therfor (9 + 1)f > a. 


IN ree Sais g> I. 

Hence g+oS¢H+. 

Hence in this casea> 98 S>¢9+ 95041. 
Therfor a>o+l, 


Moreover, since g+1 is the upper approximate 


164 INTEGERS. 


quotient obtaind by dividing a by f£,(¢+ 1)P ><a. 
S512: 
_ In both cases, since 9p Cg +1<ag<a. 

532. Theorem. /f a> 2 and f ws the integer one 
greater than the exact or lower approximate quotient ob- 
taind by dividing a by 2, theni <BP<aand BB>a. 

Let the exact or lower approximate quotient ob- 
taind by dividing a by 2 be ¢. | 

Then g++ 1 <aand (9+ 1)2> <4. § 531. 

Also g+1>1. 

Fence; if B= @ Fail =< 9 = candies x aya: 

Also, since B= 2, BBEP x2. 

Therfor BB> a. 

533. Theorem. /f a>2 and we divide a succes- 
swuly by cach number of the natural series of increasing 
positiv integers, we must after a time come to a divisor 
B, wher r<B <a, which givs an exact or lower ap- 
proximate quotient , such that 9 < f. 

Let f be the integer one greater than the exact or 
lower approximate quotient obtaind by dividing a 
by 2. 

hen wie p= Candie tseee co! 

Now divide a by f and let ¢ be the exact or lower 
approximate quotient obtaind. 


Then gfp=2a, (1) 
Now suppose g=f. 
Then gf= BB. 
Hence gf > a, 


which contradicts (5); 


DIVISION, 165 


Hence o + Band g > P. 

Therfor g<f. 

We have shown, therfor, that ther is an integer /, 
which satisfies the requirements. 


CHAPTERS Vii, 


FACTORS: 


534. We hav seen in § 487 that, if an integer has a 
given factor, it has also as a factor the opposit of the 
given factor. Moreover, no integer has zero as a fac- 
tor ($427). We will therfor in what follows confine 
our attention chiefly to positiv factors. 

535. Theorem. Zhe number of positiv factors of any 
integer, except zero,1s not greater than its numerical 
value ('). 

For we hav seen that no factor is numerically 
greater than the integer itself. 

If a is the given integer, its positiv factors must 
therfor be among the numbers 1, 2, 3, ---, |a|— 2, 
|«| —1, |@|, the number of which is | a. 

536. Hence the number of positiv factors of a given 
integer, not zero, is definit, or limited. 

537. Every integer, not zero, has unity and its own 
numerical value as factors. But not every integer 
has other positiv factors. An example of the latter 
class is the number 7, as may be seen by dividing 
7 (§505) by all the positiv integers less than it. 

538. Definition. An integer that has no positiv fac- 


(') See Tannery, p. III. 
166 


FACTORS. 167 


tor except its own numerical value, if this is positiv, 
and unity is calld a prime integer. All other integers 
ar composit integers and ar said to be factorable. 

Zero is composit and unity is prime, for zero is 
divisible by every integer except itself, and unity has 
no positiv factor except itself. 

We may determin the primeness or compositness 
of any given integer numerically greater than one by 
dividing it successivly by all the positiv integers 
numerically less than it, stopping if we come to an 
exact divisor greater than unity. 


Thus we find that I, 2, 3, 5, 7 ar prime integers 
and 4, 6, 8, 9 composit. 

539. Theorem. /f the integer a 1s prime, —a ts 
prime ; tf a 1s composit, — a 1s composit. 

For a and — a hav the same positiv factors. 

540. Theorem. //f a prime integer a ts divisible by 
an integer B, wher || +1, |a|=|8|. 

For, since @ is prime, the only positiv factors it has 
ar |a| and 1. 

Since a is divisible by f, | ?| must be a factor of a. 

Miberoresince| 2 |i-i1, 8 | == | a: 

541, Theorem. Avery composit integer, except zero, 
has a positiv prime factor numerically less than it and 
different from unity (*). 

For every composit integer a, except zero, has one 
or more positiv factors numerically less than it and 
different from unity. 





(1)See Tannery, p. 127. 


168 INTEGERS. 


The smallest of these, say 8, must be prime. For, 
if B had a positiv factor y less than it and different 
from unity, a would also hav 7 as a factor. § 460. 

Then £ would not be the smallest of the factors of 
a which are numerically less than it and different 
from unity. 

Therfor f is prime. 

542. In virtue of the last theorem, when using the 
process of § 538 to determin whether a given integer 
is prime or composit, we need not divide by any 
positiv integer that we know to be composit. 

For, if the given integer is divisible by a composit 
positiv integer, it is also divisible by the positiv prime 
factors of this composit integer. § 460. 

Hence, if not divisible by the prime factors, it is not 
divisible by the composit integer. 

543, Theorem. // » zs a composit integer different 
from zero, and aan integer such that |\v|\< aa, then v 
must hav a positiv prime factor less than | @| (*). 

Since v is composit and not zero, it has a positiv 
prime factor 8, such that 1 <8 <|»]. 

Hence v= fy, wher 1 <|7|<|»]. § Sor. 

Hence, since |v| < aa, | Br | < aa. 

Therfor either or 7 is numerically less than a. 

§ 408. 

If | 8| <|a@|, the*theorem is proved. 

In case |7|<|@]|, 7 is either prime or composit. 


Ff 





(1) See V. A. Le Besgue, ‘* Théorie des Nombres,’’ 1862, p. 48. 


FACTORS. 169 


If 7 is prime, then, since 7, and hence |7|, is’a fac- 
tor of v, the theorem is proved. 

If y is composit, it has a positiv prime factor 0, such 
PEED J 5 ogee eas ag 

Then » is also divisible by 0 (§ 460), which is less. 
than | @| (§ 193), and the theorem is proved. 

544, Theorem. // » zs an integer different from zero, 
which has no positiv prime factor less than |a|, and if 
|» | <aa, then v ts prime (’). 

This follows immediately from § 543. 

545. Formation of a Table of Prime Positiv Integers (’). 
The method of § 538 is cumbersome, if used to find 
all the prime positiv integers from I to a given large 
number. 





The following method is comparativly rapid. 

Let the given number be » and write the natural 
series of numbers I, 2, 3, ---, ». 

To determin the prime numbers in this series we 
will cross out the composit numbers. This may be 
- done by successivly crossing out the multiples, other 
than unit mutiples, of positiv integers which we know 
to be prime, other than unity. § 541. 

Now we know that 2 is prime. § 538. 

Mee usectoss out, then, the multiples of 2, 2x »2, 
CBS ORS ar Werte 

The first number after 2 of those then remaining in 
the series is 3. 





(1) See Tannery, p. 128. 
(?) See Tannery, p. 128. 


170 INTEGERS. 


Now none of the numbers left in the series and less 
than 3 x 3 is divisible by any prime positiv integer 
less: thanes except 1;. 0. 

All these numbers ar therfor prime. § 544. 

MihiUSe3ee5 e7edieprimc, 

Next we cross out the multiples of 3, 3 x 3, 3 x 4, 
3x 55 eee 

The next number after 3 in the series now left is 5. 

By the same reasoning as befor all numbers left in 
the series and less than 5 x 5 ar prime. 

Next we cross out the multiples of 5, 5 x 5,5 x 6, 
5 < Us awoke 

And so on. 

If, after crossing out the multiples of the prime 
number 7, ther remain after 7 in the series one or more 
numbers, the first of which is 0, 0 is prime and we 
next cross out the multiples 0 x 0, 0(0 + 1), (0 + 2), 
... if these ar in the series. If 0 x 0 >», the table 
is complete. All the numbers then remaining in the 
series ar prime. | 

This method of forming a table of prime numbers 
is calld the Siv of Eratosthenes. 

546. Theorem. J//, when a positiv integer v 1s divided 
successtuly by all the prime positiv integers from 2 to 0, 
wher 0 need not be prime, none of the divisions ts exact, 
and 0 givs a lower quotient » equal to or less than the 
prime number ¢ next greater than 0, the given integer 
as prime (’). 





(1) See Tannery, p. 130. 


FACTORS. r71 


To prove this, we hav » < (¢ + 1)0. § 512. 
Now, if g<e, g+IzZe, 
Hence, since 0<¢, (y+ 1)0 <ee. $340.03 50; 
Therfor y <€e, 


If g =¢, let p be the remainder corresponding to 
the quotient ¢. 


Mp chasince: Gi==so> 0 = 0, say. 

Hence, if v is divided by ¢, the lower quotient will 
ero; § 527. 

Therfor y<(d+ I)¢. 

Since 0 —€, Oetll= e. 

Hence, since gy = ¢, (0 + I)y Zee. SSE Se Ty ysis. 

Therfor p< €é, | 

In either case, then, y< ee, 


Therfor, since » has no positiv prime factor less than 
€, » is prime. 

547. Theorem. J//, when a positiv integer v 1s divided 
successiuly by all the prime positiv integers from 2 to 0, 
wher 0 need not be prime, none of the divisions is exact, 
and 0 givs a lower quotient y equal to or less than 0+ 1, 
the given integer is prime. 

This follows from § 546 as a particular case. 

For, if g20+4 1, then, sinced+12¢, gzZze. 

548. By the help of the last two theorems we may 
use the table of prime positiv integers to advantage to 
determin whether a given positiv integer v is prime or 
composit, even when » is greater than any number in 
our table. 

We divide v successivly by the prime numbers of 


172 INTEGERS. 


our table. If we get an exact quotient greater than 
unity, y is composit. If we get no exact quotient and 
come to a divisor which givs a lower quotient equal 
to or less than the next greater prime number in the 
table, the given number is prime. If we get no such 
quotient and try the last prime number 7 of our table 
as a divisor, then the given integer is prime if the 
quotient obtaind is equal to or less thany+ 1. If 
this is not so, we try 7 + I asa divisor, then 7 + 2, 
7 + 3, --- until a divisor x, less than v, givs a quo- 
tient equal to or less than « + I. 

We must come to such a divisor. For by § 533 
we must find a divisor « which givs an exact or lower 
approximate quotient less than x. 

If no quotient is exact, the given integer is prime. 
For, in dividing by all the prime numbers from 2 to 
and all the integers 7+ 1, 7 +2, 7+ 3, -+-, #, we 
divide by all the prime numbers from 2 to x. 

549. Theorem. Having given any composit positiv 
wnteger, a Set of positiv prime factors, all different from 
unity, can be found, of which rt 1s the product (’). 

Consider the number &. Dividing by 2, we find 
that © =2.x 6. Dividing the quotient 6 by 2, we 
find that G'—= 2ex73 3 Dlence sete=t 2c ce aa 

Similarly let » be any given composit positiv integer. 
Then v has some positiv prime factor less than it and 
different from unity. Let a be the first of the prime 


(1) See Tannery, p. 131; Chrystal, p. 38; Legendre, p. 5. 


FACTORS. 173 


numbers 2, 3, 5, --- by which » is divisible, the quo- 
tient being ¢,. Then »v = ag,. 

If g, is also composit and divisible by a, let ¢, be 
the quotient, so that ¢, = ay, and vy = aag,, 

Continuing this process we must come at last to a 
quotient ¢, which is not divisible by a. | 

For, let the series of quotients be 9,, ¢,, ¢,, --- 

REN) SON Os § 501. 

As each quotient is less than the preceding and all 
ar less than vy, ther cannot be more than vy — 1 of them. 
We must, then, at last come to a quotient ¢, which is 
not divisible by a. 

Then we hav » = (aaa... a times) @,. 

In the same way we may find 
g, = (PBB --- 4 times)y,, ¥, = (777 ---¢ times)¢,, etc. 

Then 
y=(aaa---a times) (P82 ---o times) (777 ---¢ times)... 

None of the quotients y,, ¥,,---, 7, is divisible by 
a, For then g, would be divisible by a. § 461. 

Similarly none of the quotients ¢,, &,,---, ¢, is 
divisible by @ or 8; and so on. 

The whole series of quotients ¢,, %,, +++, Gi Yi» Yo 

yyy) oa, Vo must, Navan. end, the, last 

quotient being a prime number different from unity, 
since, for the same reasons as given above, ther ar not 
more than vy — 1 quotients in all. 

We hav, then, found a set of positiv prime factors, 
of which » is the product. 

550. Definition. The process described above for 


174 INTEGERS. 


finding a set of positiv prime factors different from 
unity, of which a given composit positiv integer is the 
product, is calld resolution into prime factors. 

In this sense we cannot properly speak of resolving 
a prime positiv integer into prime factors. We may, 
however, regard it as already resolvd. 

551. Definition. A common factor of two or more 
integers in an integer that is a factor of each of them. 

Thus 3 is a common factor of 6 and 9; 5 is a com- 
mon factor of 5 and +. 

552. Theorem. Zhe number of positiv common fac- 
tors of two or more integers, of which at least one is not 
sero, 1s not greater than the numerical value of the 
numerically smallest of those which ar not zero('). 

Let the number of positiv common factors be z and 
let a be the numerically smallest of the integers which 


ar not zero. 
Then, since the z positiv common factors ar factors 
of a, 7 > |a|. S20 sihe 


553. Theorem. Thus ¢he number of positiv common 
Jactors of two or more integers, of which at least one is 
not zero, ts limited. 

554, Theorem. Zhe number of positiv common fac- 
tors of two or more zero integers 1s unlimited. 

555. Definition. An integer a is prime (’) relativly 
to, or simply prime to, an integer 8, when a and # hav 
no positiv common factor except unity, 





(1) See Tannery, p. 112, 
(?)See Chrystal, p. 38, 


af 


FACTORS. IAs 


Thus 4 and 9 ar prime to each other. 

The statement ‘a is prime to #”’ may be written 
a||; the statement ‘a is not prime to ®” may be 
written a /f f. 

556. Theorem. No integer except positiv and negativ 
unity ts prime to ttself. 

557. Theorem. Jf a || f, chen B || a. 

558. Theorem. Jf a }{ 8, then B lh a. 

559. Theorem. /fa=f and B || 7, then a ||7. 

560. Theorem. /fa|| 8 and B=y7, then a || 7. 

561. Theorem. Jf «||, then a|| P, «|| B, and 
a || B. s ; 

For a has the same factors as a, and f as P. 

562. Theorem. a || I. 

563. Theorem. The only positiv integer to which zero 
1s prime ts unity. 

564. Theorem. <Axy two different prime positiv in- 
tegers ar prime to each other. 

565. Theorem. J// « || P, every factor of a ts prime 
to every factor of (°). 

For, if these factors had a positiv common factor 
other than unity, a and # would hav this factor also. 

566. Theorem. /fa>> 8 wher | 8| + 1, thena ff P. 

For, if a >> 8, wher | #| + 1, @ and Pf hav a pos- 
itiv common factor, | f |, different from unity. 

Therfor a /{ f. 

567. Theorem. /fa || 8, wher|8| + 1, thena ts p. 

This follows immediately from the preceding. 


(1) See Tannery, p. 117. 


176 INTEGERS. 


568. Theorem. /f a}{ f, wher B is prime, then 
a> Pp. | 

By the hypothesis P + o. 

For zero is a composit integer. 

Moreover | ?| + 1. 

For every integer is prime to unity. 

Now, since a }{ 8, a andf must hav some positiv 
common factor besides unity. 

But the only positiv factors of # ar 1 and | /|. 

Hence: alee ia 

Therfor a> P. 

569. Theorem. /f af, wher 8 ts prime, then 
as) (e8, 

570. Theorem. /f |a|>| |, wher a is prime and 
B + 0, then a \} Pp. 

Since @ is prime, the only positiv factors it has ar 
|a| and 1. 

Moreover, no positiv common factor of a and can 
be numerically greater than /. 

Hence unity is the only positiv common factor of a 
and 

Therfor a || f. 

571. Theorem. /f a= 98+ 7, any common factor 
of Band 7 ts a common factor of a and 8, and con- 
versely ('). 

The direct theorem is proved easily from § 480. 
The converse follows from § 481, since, ifa= 98 +7, 
T= OR 

(1) See Tannery, p. 112; Chrystal, p. 39. 


‘CHAPTER VIII. 


GREATEST COMMON FACTOR. 


572. Definition. We hav seen that the number of 
positiv common factors of two or more integers, of 
which at least one is not zero, is limited (§ 553).. 
Among these factors ther is one, and only one, which 
is greater than all the others (§ 199). This is calld 


their greatest common factor ('). 
It may be found by dividing each of the other in- 


tegers by the factors of the numerically smallest of 
those which ar not zero. 

Thus the greatest common factor of 8 and & is 4. 

Two or more zero integers hav every positiv integer 
as a common factor and hence hav no greatest com- 
mon factor. 

We will denote the greatest common factor of two 
integers a« and f, which ar not both zero, by the 
symbol a, which may be read “a kor f.” 

The symbol 4@/ will, then, be univalent, if a and 
8 ar not both zero. The symbol 0 @o, however, 
will be meaningless. 

The integers a and # may be calld the elements of 
the greatest common factor 4@ . 


(1) See Tannery, p. 112. 
12 177 


178 INTEGERS. 


573. Operation of Finding Greatest Common Factor. 
In § 572 a method for finding the greatest common 
factor of two or inore integers, which ar not both zero, 
was explaind. The greatest common factor 4 is 
therfor found by an operation. 

574. Theorem. a®&f= 8a, provided a and f 
ar not both zero. | 

This follows immediately from §§ 553, 199, 170. 

This theorem is the Commutativ Law for Finding 
Greatest Common Factor. 

575. Theorem. /far>> 8, a&WP=| P| (’). 

For the greatest positiv factor of f is | |. 

And, since a >> , | #| is also a factor of a. 

Therfor | f | is the greatest common factor of a and /. 

576. Theorem. a@a=|a|, provided a + o. 

For then a > a. § 442. 

577. Theorem. a@®I= I. 

578. Theorem. OWa=| «|, provided a + oO. 

579. Theorem. J/f all f, a&WB= 1, and con- 
versely. 

580. Theorem. 

(— 4) @8=4@(-8)=(—Y@O(—- 8) = 48, 
provided a and f ar not both zero. 

581. Theorem. //|7| 2s unity, a Wy =P Oy. 

For then both a@; and §@7 ar unity. 

582. Theorem. // |7| zs unity, rWa=7 WP. 

583. Theorem. /f a=, aWr=— Wy, provided 
B and y ar not both zero. 





(1) See J. A. Serret, ‘‘Traité d’ Arithmetique,’’ 1852, p. 79. 


GREATEST COMMON FACTOR. 179 


For, if a and # ar equal, they ar the same integer. 

Hence they hav the same greatest common factor 
with 7. 

584. Theorem. /fo=8, ,Wa=7W8, provided 
B and y ar not both zero. 

585. Theorem. /fa= Pandy =0,4@r= RO, 
provided 8 and x ar not both zero. 

586. Theorem. /fa= 98+ 7, thnaW P= Wy, 
provided a and 8 ar not both zero(’). 

For a4@/ isa common factor of 8 and 
and #@jy is a common factor of « and f. § 571. 

Hence, if c&®P > Wr, B&yr would not be the 
greatest common factor of # and 7. 

And, if «@P <PWr, «& would not be the 
greatest common factor of a and f. 

heron GO e — 2 6O7. 

587. Theorem. /fa=o8+y7 anda || f, then P || 7, 
and conversely. 

588. Theorem. 1 + af zs prime to a and f. 

For a and f are prime to I. 

589. Theorem. 1 + afy---2s prime toa, 8,7, --- 

590. Theorem. Having given any number of prime 
positiv integers, all different from unity, another can be 


found. 

For, let a, 8, y,---, « be any series of prime positiv 
integers, all different from unity, 
and let A= afy-+-«+ 1. 





(1) See Le Besgue, p. 32; Chrystal, p. 39. 


180 INTEGERS. 


Then A> 1 and A is not divisible by any of the 
integers a, B, 7, +++, X. §§ 589, 567. 

Now A is either prime or composit. 

If 2 is prime, the theorem is proved. For, since A 
is not divisible by any of the integers a, f, 7, ---, «, it 


cannot equal any of them. § 447. 
If A is composit, resolv it into positiv prime fac- 
tors. § 540. 


These factors must all be different from the inte- 
gers a, 8, 7, +++, *, since A is not divisible by any of 
these integers. 

Therfor another prime positiv integer, different from 
unity, can be found. 

591. Theorem. Zhe number of prime positiv integers 
is infinit('). 

592. Theorem. Having given any prime positiv inte- 
ger, a greater prime positiv integer can be found. 

Let « be any prime positiv integer. 

From the natural series of numbers 2, 3, 4, --:, « 
pick ‘out the primesintesers)) Wet) theses begs 
e ogGas | 

Then, by § 590, having given the series of prime 
positiv integers a, 8, 7, ---, «, all different from unity, 
another can be found. 

But this other cannot be less than «, since all the 
prime positiv integers different from unity and less 
than « ar included in the series a, f, 7, -+:, «. 





(1)See Euclid’s Elements, Bk. IX, Prop. 20; Chrystal, p. 47; 
Tannery, p. 127. 


GREATEST COMMON FACTOR. I8I 


Therfor this other prime positiv integer is greater 
than x. 

This theorem is equivalent to the statement that 
ther is no greatest prime positiv integer. 

593. Theorem. Having given two integers a and f, 
wher 2 +0, of we divide a by B, calling the remainder 
p+ then, of p, += 0, divide 8 by p,, calling the remainder 
Px then, if p, + 0, divide p, by p,, calling the remainder 
P,* and so on, that ts, uf we divide each remainder by 
the succeeding remainder, provided that succeeding re- 
mainder 1s not zero, we must finally come to a zero 
remainder. 

For, by § 519, |2|> ules fee ee Rep eerers 

Hence the remainders continually decrease in 
numerical value and ar all numerically less than f. 

The number of remainders cannot, then, be greater 
than the number of integers in the series |#|— 1, 
|2|— 2, |8|—3, --+ 3, 2, 1, 0, which is | @]. 

We must, therfor, at least as early as in the | ?|’th 
division, find the remainder zero. 

594. Theorem. Having given two integers a and f, 
wher B+0; if we dwide u by 8, calling the remainder 
p,» then 8 by p,, calling the remainder p,; then p,, by 
Px, calling the remainder p,; and so on, until we come 
to the zero remainder ; the greatest common factor of a 
and B ts the numerical value of the divisor that givs the 
sero remainder. 

Pammtieszeronremamict, 0. usct, a —i9 1, 8 == 0), 
and call the successiv quotients 9,, 2,, Gy +++, Q,. 


182 INTEGERS. 


From the series of divisions we get the series of 
equalities CF at 9,8 T Py OF P_) =, Py + Py 
B= 5), + Py Po = Po Pr + Pe 
Cite Bt an fs = Hf f th He 


Cae ea ye ee ae ele yd aabes 


cB oe 
Hence 4@Q 2 = 8 OA = 1 OP = P20 Ps 
= Pro © Pat = Pat O Pn ‘s 586. 
=P, © 0 § 584. 
=|?,-1|- § 578. 
This method for finding the greatest common fac- 
tor of two integers is calld the Algorithm of Euclid (’). 


595. Theorem. //a || f, | p,_,| = 1, and conversely. 


596. As the greatest common factor of two integers 
is an integer, we may combine it with other integers 
by any of the signs +, —, x, :, ®, provided that 
the first element of every quotient is divisible by the 
second, and that of two integers connected by the 
sign @ not both ar zero. We will use parentheses 
to indicate the order in which the various operations 
ar performd. 

Every such expression is univalent. For every 
sum, difference, product, quotient, and greatest com- 
mon factor is univalent. ° 


(1) See Euclid, Bk. VII, Prop. 2; Chrystal, p. 39; Tannery, p. 
113; Dirichlet- Dedekind, p. 6. 


GREATEST COMMON FACTOR. 183 


597. Theorem. Jf for one or more elements of a 
complex expression which contains integers connected by 
auy or all of the signs +, —, X, %, &, and in which 
the first element of every quotient is divisible by the 
second and of two integers connected by the sign (® not 
both ar zero, equal integers ar substituted, the Baggies 
expression ts unchanged. 

598. Theorem. 

If a + 0 and 8 + 0, (af) @ (7) =|4|(8 @7)(. 

To obtain 6@&y we proceed by the algorithm of 
Euclid, obtaining the remainders /,, (,, Ps, +++, P,—» 
Se § 504. 

iitien oe =O ander Co je— 1 aan) 

To obtain (af) ®& (ay) we proceed in the same way. 

But by § 524 a possible series of remainders will be 


OP Pay EDs FP py EP iy FP: 


PANGesINCclen—— 0, COn— O, 
Therfor (43) © (47) = |a?,4| =| 41] P| 
=|2| (2&7). 

This theorem is the Left-handed Distributiv Law for 
Multiplication and the Operation of Finding Greatest 
Common Factor. 

599. Theorem. 

If 8 + o andy + 0, (a7) @ (Fr) = (@@A) |r|. 

This theorem is the Right-handed Distributiv Law 
for Multiplication and the Operation of Finding Greatest 
Common Factor. 





(1) See Tannery, p. 116. 


184 INTEGERS. 


600. The preceding two theorems ar frequently of 
help in obtaining the greatest common factor of two 
given integers. Any common factor that is known to 
exist may be removed and the greatest common 
factor of the remaining numbers found. If this great- 
est common factor be multiplied by the numerical 
value of the factor that was removed, the product 
will be the required greatest common factor. 

601. Theorem. /f a+0, B+0, and B\\7, then 
(48) & (ay) = | a|; and conversely, if (oP) (az) = |4|, 
then B || +. 

For the direct theorem, since P || 7, BW7 = 1. 

Hence (ap) G0 (ay) <u Gxt a § 598. 

For the converse theorem, 
we hav (a8)@ (a7) = || (PW). § 508. 

But by the hypothesis (af) @® (ay) = | a|. 

Hence [a1 (8 Go7) —=saieand: 2 60 yar. 

Therfor pepe 

602. Theorem. Jf ary and B77, wher P+ 0, 
then «@ > 7 and (a@8) :\7| = (4: Q(B en) 0). 

We hav a= gy and 8 = yy, wher y + o. 

Hence «@P= (7) @ U7) = (#QVI|r| 

$$ 585, 590. 

Therfor a@8 >7 
and (2@A)?|rl= 9 @xr=(@:N@OE:7). 

This theorem is the Distributiv Law for Division 
and the Operation of Finding Greatest Common Facter 





(1) See Le Besgue, ‘p. 32. 


GREATEST COMMON FACTOR. 185 


603. Theorem. /f B-+0,; then, if |0|=aWBA, 
a6, B30, and a:6\|| 8:0; and conversely, of 
QO, 8 > 0, anda: \\P: 0, then || = a6dp. 

For the direct theorem, 

nO ead) (pe) 0) = Te S002: 
ences | [7202 0: 


For the converse theorem reverse the steps. 


604, This theorem is equivalent to the follow- 
ing : 

Theorem. Jf a and § ar two integers, which ar not 
both zero; then, if « is thew greatest common factor, 
ther exist two integers y and 0, prime to each other, such 
thata = 7x and B = 0x, and conversely, if a= Kn and 
8 = 0x, wher 7 and 0 ar prime to cach other and x ts 
positiv and 0 not zero, then kts the greatest common fac- 


tor ofa and fi. 


605. Theorem. Zhe common factors of two integers, 
which ar not both zero, ar the same as the factors of 
ther greatest common factor. 

For, by §§ 572, 460, every factor of the greatest 
common factor is a factor of each of the given in- 
tegers. 

And, by § 602, every common factor of the two 
given integers is a factor of their greatest common 
factor. 

Hence the two groups of factors, first, the common 
factors of the two integers and, second, the factors of 
their greatest common factor, ar the same. 


186 INTEGERS. 


606. Theorem. Zhe integers a and 8, wher B + 0, 
and all the remainders obtaind in finding their greatest 
common factor by the algorithm of Euclid can be exprest 


in the form ha + pf’). 


We hav P= Filo FT Pv 
Po = $21 + Px» 


Pica P sls ares 
Pear Cle aia tgs 
wher ,, is the last remainder, the zero remainder. 


§ 594. 
Now a= I.a— 0.8, 


or p_,=A_a+ p_,f, wher 1_, = 1 and p_, = — 0. 
Again 8=—0.a+41.f, or pp= 40+ 48, 
wher pee) NE Roi, 
Thus the first two of the numbers a, f, (,, A Pa °° 
p,, hav been exprest in the desired form. 


*y 


Suppose we hav proved that two successiv p’s, p,,_, 
and p,_,, wher m— 25 — 1 and m— 1=nz—1, that 
is, wher O< m<u+1, can be written in the form 
ha. aP LB, sO that Pao aa Aa =. (Pie 
and Pm—-1 Kay ae ar bn iP 
Then, since Cn ea aes tg Pn? 
by substituting in this equality the values of p,_, and 
P—2 taken from the preceding two equalities, we get " 





(1) See Chrystal, p. 44. 


GREATEST COMMON FACTOR. 187 


Cn foo 2 =f Ln) ar (A, 2 ay [Perpagteh 
a (— Ge AES aF his ag) te = Pernt an Um —2)2 


=< ha +. Tjek 
wher i, a Ge hee. ae User 
and Pr ete Cl =r Pn—2 


Thus we hav proved that, if two successiv remain- 
ders p,,_, and p,,_, can be written in the form Aa + py, 
the next remainder p,, can also be written in this form. 

But we know that p_, and p, can be written in the 
form 4a + yf. Hence p, can be written in this form. 
Again, since p, and p, can be written in this form, p, 
canbe. Andsoon. That is, all the p’s can be written 
in this form. Therfor p, = 4,4 + yf, where may hav 
any value from — I to x. 


The formulas 2, = —¢ 4, +4,_. 
and bm = — Pm ima Tt Ema 
show how, having given A_,,/, and /_,, 4, and the series 
of ~’s, the remaining /’s and p’s can be calculated. 
Thus 4,=—9A,+4_) y= — Pio + Pp 
, emi cea M2 Te Sen “aly 


It will be noticed that the formulas 5 which 4, and 
yp, ar calculated from 4,_,, 4, and /4, 4) fn—2 respec 
tivly ar similar to that by which p,, is obtaind from 
eet and mi=9" 

607. Definition. The method of proof just used to 


m—1) 


188 INTEGERS. 


show that p, can be written in the form /,a + 4,8 for 
all possible values of ¢ is calld the Method of Mathe- 
matical Induction. 

According to this method we prove that if a certain 
rule, in whose statement ther appears an integer 7 
holds for a particular value of 7, or for some specified 
number of successiv particular values, then it holds 
for the next value. 

Then we show that it does hold for a particular 
value of 7, or for the specified number of successiv 
particular values. 

Hence it holds for the next value and therfor for all 
values that 7 can take, starting with and succeeding 
some particular value. 

This result may be stated in the form of a theorem 
as follows : 

Theorem. Law of Mathematical Induction. Jf a 
certain statement, in which an integer n is involud, 
holds for a particular value of 4 or for some spect- 
fied number of successtv particular values, and, if it 
is proved that, if the statement is true for a particular 
value of 4 or for the specified number of successiv par- 
ticular values, wt holds for the next value, then the state- 
ment is true for all values that 4 can take, starting with 
and succeeding some particular value. 

This is so, because every integer has its place in 
the natural series of integers. 

608. By the method of § 606 we can calculate as 


Ff 


GREATEST COMMON FACTOR. 189 


many values as we please of the 4’s and p’s in terms 
of the ¢g’s, up to 4, and yz. 
Thus we hav 


1 = el iis aga tOP 
4,=—9, 4H = +1, 
A=+1, Paar at 

1, = — ¢» Py = + (9.9, + 1), 
A 


3= + (3%, + 1), Ps = — (P3201 + 3 + ¥1)s 

It will be noticed that the signs with which the 
values of the 2’s and p’s begin ar alternately plus and 
minus and that each A has the opposit sign to the cor- 
responding yp. This rule holds for all the /’s and p’s 

For, sinceA, = — 9,4, _, +4,_,, if 4,_, has the plus 
sign and A, _, the minus sign, 4, will hav the plus sign 
($326). And if2,_, has the minus sign and /,_, the 
plus sign, 4, will hav the minus sign. 

Hence, since A_, has the plus sign and 4, the minus 
sign, A, has the plus sign; since A, has the minus sign 
_and A, the plus sign, 4, has the minus sign ; and so on. 

Similarly for the p’s. 

Also, since 4_, has the opposit sign to #_,, and the 
signs of the 4’s and p’s alternate as we go down the 
series, the signs of the corresponding /’s and p’s will 
continue to be opposit. 

609. Theorem. /f 4,_,,A, and p_,, pt, ar any par 
of successiv i's and the corresponding p's in the series 
Weert A eA ONd Ue fy, fn fk given in § 600, 
then h, pp, —A py, = £ 1, according as ets even or odd, 


190 INTEGERS. 


Set 7, =A, —A,4,_, wher —I <e<u4+t. 
Then ip = ih JP 
=IxXI—OxX OI. 

Now we hav 4, = —@A,_,+4,_, 
and [iO at 2, 
which equalities hold wheno<e¢<u+1. 

By multiplying the second of these equalities by 
A,_, and the first by #,_,, and subtracting, we get 


Ae rT} Ae i sus, ay AvaiLe) 
or T, = —T,_,, which holds ifo<e<u+1. 


In this equality replacing « by ¢— 1, we get 


T,_, = —T,_,, which holds if oc¢e—1<+1, or 
iC Saat ee eh n+ Pap 
Pence ec a Otac ee oe WICH Old omwiter 


both o°=< 7< 2-4 rand 1—#= 7” + 2,'thatus, when 
) Geet a JE ah 

In the latter equality giving ¢ the values 2, 4, 6,-.-- 
iNPSUCCESSION yawersec tates ca eet hens 


Giving ¢ the values 3, 5, 7,--- in succession, we 
get T, =T,=7,=T,=-:- 

Now t, ==. ences a s—a1 whenever ecmisman 
even number greater than — 1 and less than z 4 1. 

Also, since Tt, = —T,_,, wheno <¢< “+1, 

Ret Nell (Yat mere, 118 

Hence t,=— 1, whenever ¢ is an odd number 

greater than — 1 and less than z + 1. “ 


610. Theorem. // a and 8 ar two integers, which ar 


GREATEST COMMON FACTOR. IOI 


not both zero, two integers hand pcan be found, such 
that ha + p38 =a FB. 

Ryesavea (8 —i bor. 
And Puma = At + By a8. 
Hence 

| Pot | = $4 + Pf) = £4, @ + p,_B, 

the upper or the lower sign being used according as 
Pa— 

If, then, p,_, is positiv, 
we hav Ame iE [Leen OS) 

If p,_, is negativ, 

—i _ja—p_P=a@f. 


In the first case setting 2,_,, 4,_, respectivly equal 


, 1S positiv or negativ. 


to 4, w and in the second case setting 4,_,, 4,_, equal 
to — A, — yw, we hav for each case da + pf? = af. 

611. Theorem. //f a and fP ar two integers prime to 
each other, two integers h and pw can be found, such 
that ha + pp —— Whe 

According to the hypothesis a and # cannot both 
_ be zero, because zero is not prime to itself. $556. 

Hence this theorem follows from the previous one 
and § 570. 

612. Theorem. /fa+ P=1, then-a || P. 

For, since a+ # = 1, any common factor of a and 
f is a factor of I. § 478. 

613. Theorem. /fa+t B=1 andy and 0 ar factors 
of aand B respectively, 7 \\ 0. 

This follows easily from the preceding theorem and 
§ 460. 


192 INTEGERS. 


614. Theorem. Jf 2a + pf =1, each of the integers 
1, ats prime to each of the integers p, P. 

This follows from the preceding theorem, since 
TGP ETD Pet or eh (AKO 

615. Theorem. Jn the series h_,, 4, 4, +++, 4, and 
Poa» Py Py tts Py Stven in § 606, each dis prime to 
the succeeding h, cach p to the succeeding p, and each i 
to the corresponding ft. 

This follows easily from the preceding theorem and 
§ 609. 

616. Theorem. //ha+ pP =aWFB, then 2 || p. 

We hav a= j7« and 8 =dx, where =a@f. § 604. 

Hence lyk + poK = kK. 

Hence Ay + po'= I. 

Therfor Ali] yw. § 614. 

This theorem might be stated. Jn whatever way 
af ts exprest in the form ha + pB, A and ar prime 
to cach other. | 

617. The most important particular case of the 
preceding investigation, §§ 593-611, is the case when 
a and f ar positiv. Let us consider this case. 

Suppose a> fand that the quotients @,, ¢,, ---, ¢, 
ar all lower approximate quotients, except y, which 
is exact, so that the remainders 9p,, ¢,, ---, e, ar all 
positiv, except p,, which is zero. 

Then the greatest common factor of a@ and f is 
,,-y Which is unity, if 2 and # ar prime to each other. 

The quotients ¢,, @,, ---, g, ar all positiv. § 526. 

Hence the values of 4,, 4,, ---, 4, (§ 608) ar alter- 


GREATEST COMMON FACTOR. 193 


nately positiv and negativ, the positiv values corre- 
sponding to odd subscripts. 

The values of 4, 4, ---, #, ar also alternately 
positiv and negativ, the positiv values corresponding 
to even subscripts. 

Of any A and the corresponding 4, one is positiv 
and the other negativ. 

Let us now put 4,, A, 4,, --- respectivly equal to 
Viy Yay Ysy +275 Ay Ay Agy +++ Equal to —yv,, —v,, —v,, 

SEY dig Def ee ag af emote equal (Oe S, sat. sche a hee 
Pee equa letane. S776... 

Then all the v’s and &’s will be positiv and we will 

hav 


Minar of; fF. =, 

ay eas ae f= 9,9, + I, 

v= 99, + I, = = $392, + P3 + Py 
Py = ¥4a— 6&8, 
Py = — v4 + &,8, 


em Tae Ef, 


a. ae. 
Pn» = = (v4 — &,,2), according as 
m is odd or even. 

Also Viegas Oe Viento Dae) 

ees Cate tee weet lil git o> 2: 

For, if m is odd, the formulas, = —g A, +4,_, 
givs PAE ES etfs Say a as 
or Dae ead ai 

If 7 is even, the same formula givs 


13 


194. INTEGERS. 


— Y= — OY na — Yn 
or Se Es. 
the same as in the other case. 

Similarly the formula for €, is proved. 

We can now obtain from the theorems of §§ 606, 
609-611 the following theorems for this particular 
case. 

618. Theorem. /f aand 8 ar positiv integers and 
a>, all the remainders obtaind in finding their 
greatest common factor by the algorithm of Euchd can 
be exprest in the form + (va — &8), wher v and & ar pos- 
ativ integers, the upper sign being used for the odd re- 
mainders, the lower for the even remainders. 

619. Theorem. Jf», », and& _.,€, ar any pair 
of successiv vs and the corresponding §’s in the series 
Yi Dany Ve OU emer, ee, ek ee 
according as m is even or odd. 

For, if 7 is even, the equality A, _¥4,—4,4,.= +1 
givs Deore ae v= —) prog 8: 

“ys eat rare Eg pubs ones bs ie 

Similarly the other case is proved. 

620. Theorem. x the series v,, v,,-+-, v, and &,, &., 
-+& each vis prime to the succeeding v, each & to the 
succeeding §, and each y to the corresponding §. 

621. Theorem. Jf a and £8 ar positiv integers, two 
positiv integers v and § can be found, such that 


va —§8 = +a@B(). 


(1) See Le Besgue, p. 37; Chrystal, p. 45. 


GREATEST COMMON FACTOR. 195 


622. Theorem. Jf a and £ ar positiv integers prime 
to cach other, two positiv integers v and € can be found, 
such that va — EB = +1. 

623. We will now leave this particular case and 
continue the general investigation. 

Theorem. /f Ga >> 8, wher a|| 8, then 0 > 8(’). 

Since a|] 8, we can find A and y, such that 

1 = at pf. SfOLI 

Multiplying by 0, 0 = Ada + pO. 

Sines (ieecyeh Mie ile 

Substituting gf for #a in the preceding equality, 

0 = 1y8 + w03 = (iy + pO). 

bihenotemdes 7: 

624. Theorem. /f a|| 2, any common factor of Oa 
and 3 must be a factor of 6 (’). 

625. Theorem. // 6|| 8, (c0)\®WP= 4B, provided 
a and 8 ar not both zero (*). 

For a@®f is a common factor of af and 8. § 460. 

And (a0) @ fis a common factor of aand 8. § 624. 

Hence, if (A)®BP>a@f, «@Pe would not be 
the greatest common factor of a and f. 


And, if (@)\®P <a@F8, (44)& PF would not be 


the greatest common factor of a and f. 


Therfor (af)®Q B= af. 





(1) See Chrystal, p. 41; Tannery, p. 119; Lucas, p. 339. This 
theorem was known to Euclid. 

(2) See Dirichlet-Dedekind, p. 8. 

(#) See Tannery, p. 118. 


196 INTEGERS. 


626. Theorem. Jf ar>>@ and O6\|8, then 
(2: 0)\@Q8=4Q8, provided « and 8 ar not both zero, 


Since a > 6, a= 9, 

Hence aQ@Q P= (¢0™)\&P § 583. 
=¢&P § 625. 
i CTEM C8) es: § 583. 


627. Theorem. (a)®&y zs divisible by BW yz. 
For we hav 2 = 0« and 7 = ex, wher k= 8. 


§ 604. 
Hence (48) @7 = (40x) & (ex) = [(40) & e] x. 

§ 599. 
Therfor (42) @7 >> «. § 409. 


Or = (4) Wr > F Wr. 
628. Theorem. // y> da and y > OB, 


wher ytoOo and al|\f8, then v>> Gap 
and |v| :|0aB| =(: (02)) OY: (68); 

and conversely, if v >> Jaf, then v >> Oa, » >> OB, and, 
if \v|:|@aB|=(: (2))WC: (6f)), then a || B. 


For the direct theorem, 


we hav y= 90a andy = y6p. 
Hence gba = 08. 
Hence ga = xf. § 339. 
That is, ga > p. 
Hence, since a i, ort Pp. § 623. 
That is, piesAy, 
Hence y= Ada. 


Therfor v >> 6a and v: (G@a8)=A, v: (Ga) =A, 
v3 (08) = Aa. 


GREATEST COMMON FACTOR. 197 


Hence |»|:|@a8| =| 2| 
and 


(» : (0a)) OY : (OB) = a & (Aa) wee 
= Ol. 
Therfor |» |: | @a2| = (v: (Aa)) @Q(v : (48). 


For the converse theorem reverse the steps of the 
latter part of the proof. | 

629. Theorem. /fv >a and» > 8, whery + oand 
all B, then v> a8 and |v|:|a8| =: a) QO: A); 
and conversely, if v>> ap, then v>>a,y>P, and, 
if|v|:|48|=:4)@: P), tena || £. 

This theorem is a particular case of the preceding 
theorem, the case when 0 = I. 

The theorem of § 579 is a particular case of this 
theorem, the case when v = af. 

630. Theorem. // y>>-a and v>BP, wher 
pe O, then vy > (a8) : (4@&F) and 
ly] : (a8 | (@@B) =) Ql: 8). 

We hav ik Cs ==" 10: 


wher a@P=« and;z IL 0. § 604. 
mince YS aandy > 8) ys} xy and y >} «0, § 450. 
Hence yp > Kyo 

and |v|:|«yd|=(¥: («7)) WC: (Kd). § 628. 
Moreover af = (x7) (KO) = (Ky0)k. §§ 331, 381. 
Therfor af >> « and (a8): «= kyo. § 414. 


Hence |«yd|=|(a8):«|=|aP|:«  §§ 396, 406. 
= ||: (4@8). $459. 

Therfor v>>(a@8):« and 
ly] = (47 | = ZAP) =% 24) @OU:F) $8459, 585. 


198 INTEGERS. 


631. Theorem. /f a || 8 and a || 7, then a || Pr; 
and conversely, if a \\ By, then a\\ 8 and a\\7(°). 


For the direct theorem, since a || P, a and f can- 
not both be zero. 


Hence, since a |'[97 9a O07) — 2 00us § 622. 


=I, § 570. 
Therfor here § 579. 
For the converse theorem, 
since a || £7, a. & (87) att 


Also a and f cannot both be zero. 

For then @ and fy would be zero, which cannot be, 
Since a] | 87. 

But 4 (7) is a multiple of both a@f and aj. 

§ 627. 

Hence, if either a@P or a®&y were not unity, 
a(@& (fr) would not be unity. 

Therfor both a@ and a4@ ar unity. 

‘Thertor a |} 8 andia)| 75 

632. Theorem. // ais prime to cach of two or more 
integers 8, 7, 0, +++, i is prime to their product Byd ---; 
and conversely, if ais prime to the product Byd-+-, then 
wt 1s prime to each of those integers (*). 

We hav already proved the theorem for the case of 
two integers. 

Suppose ther ar more than two. 

Then, for the direct theorem, we hav first a || fy. 

S021 
(1) See Tannery, p. 119. 
(2) See Chrystal, p. 41. 


GREATEST COMMON FACTOR. 199 


Hence, since a || fy anda|| 0, a || (Br)d. § 631. 


And so on. 

For the converse theorem, since @ || S(y0--;), 
a||Pandal|lj7o..-. § 631. 

Similarly @ can be proved prime to each of the 
other integers. 

633. Theorem. /f cach of the integers a, 8, y, -++ts 
prime to each of the integers 1, p,v,---, the product 
afy-.-7s prime to the producthyy...; and conversely, 
if the product aby ..-1s prime to the product hy ..., then 
each of the integers a, 8, 7,--- 1s prime to each of the 
integers h, p,v---("). 

For the direct theorem, we hav first a [| Ay... 


§ 632. 
Similarly P [| Aw... 
And so on. 
Therfor aBy.-- || Apy--- § 632. 


For the converse theorem, since afy--- || Aw-.- 

we hav a || dAwy.--, Bll Aw.--- 7 |p Aw---, >> 
§ 632. 

Since a | [Age as|ileAa leu a flys. 

Similarlye9,7,-.-ar prime to A, 4, vy, - +. 

634. Theorem. /f a || [, the product of any number 
of a's ts prime to the product of any number of fs ; and 
conversely, if the product of a number of a's ts prime to 
the product of a number of f?'s, then a \\ P. 


This theorem is a particular case of the preceding 


(1) See Dirichlet-Dedekind, p. 9. 


200 INTEGERS. 


theorem, the case when a, f, 7, --- ar the same integer 
and 2, p, v,--- ar also the same. 

635. Theorem. Jf the product aBy ..- of two or more 
integers ts divisible by a prime integer i, one of the fac- 
tors a, 8, 7, +++ must be divisible by i(°). 

For, if none of the factors were divisible by /, then 
A+ 1 and these factors would all be prime to 2. 


S$ 443, 569. 
Hence the product af; --- would be prime to A. 


$$ 557, 632. 

Therfor this product would not be divisible by A 
(§ 567), which contradicts the hypothesis. 

636. Theorem. Jf the products of two sets of prime 
positiv integers, no one of whichis unity, ar equal, the 
two sets ar the same, except that the factors may be aif- 
JSerently arranged in the two sets, 

Leta, 8) 7) andes a bere twa sets, 

Then, since afy .-- = Au... and Aw ... is divisible 
by A ($$ 381, 453), @Sr--- is divisible by 2. § 458. 


Hence sone) of the intesers apo) os asaya 


divisible by A. § 635. 
But a and 4 ar primes and 4 + 1. 
Hence a=, § 540. 
Therfor Byr-++= py--> §8§ 381, 341. 
Similarly ath 


And so forth. 
Now ther must be the same number of integers in 
the two sets. 


(1) See Tannery, p. 131. 


GREATEST COMMON FACTOR. 201 


For, suppose ther were not the same number. 
By applying the above method we would, then, 
come to an equality of the form 


€ = Pot::-, or (.1 = pot..-- 


By the same method as befor, we would now get 
I= oT-:- 

Hence 1 would be divisible by oa, an integer greater 
than 1, which is impossible. 

Therfor the numbers of integers in the two sets 
must be the same, and the sets ar identical, except 
possibly for the order of their factors. 

This theorem is usually briefly stated: 

A positiv integer can be resolud into positiv prime 
factors in only one way (°). 

637. The representation of numbers by products 
of prime factors is a very useful artifice in many prob- 
lems. Various operations ar renderd easier by this 
means and many properties of numbers ar made more 
evident (°). 

We will giv a few illustrations. 

638. Theorem. // two or more positiv integers ar re- 
solud into prime factors, their product, resolvd into 
prime factors, will contain every distinct prime factor 
that 1s in either of the given numbers ; of any prime fac- 
tor occurs in only one of the given numbers, it will oc- 





(1) See K. F. Gauss, ‘‘ Disquisitiones Arithmeticz,’’ 1801, p. 9; 
Chrystal, p. 44; Tannery, p. 132. 
(2) See Tannery, p. 134. 


202 INTEGERS. 


cur the same number of times as a factor in the prod- 
uct; if any prime factor occurs in two or more of the 
given numbers, the number of times tt will occur as a 
factor in the product will be the-sum of the numbers of 
times that it occurs as a factor in these numbers |’). 

Let the given positiv integers be »,, v,, Ys, +++ 

When resolvd into prime factors suppose 
y,=(aaa--- a, times)(fP8--- 3, times) (777 +--+ ¢, times). - - 
v= (aaa .--.a,times)(d00 --.d,times)--., 
y,=(aaa---a,times)(777 --+c, times) --., 


The product of the given integers may then be 
written : 

(aaa---a, times)(f88 -.-d, times)(y77 +--+ ¢, times) --- 
x (aaa.--a,times)(d00 ...d, times) - - - 
x (aaa--.a, times)(777 +++ Cc, times) --- 
57 ok ora | age Gy eee Sem eer eee: § 384. 

Since all the factors here written ar prime, this ex- 
pression is the same as the expression for the product 
of the given numbers when it is resolvd into prime 
factors. § 636. 

From its form it is evident that this expression con- 
tains every distinct prime factor that is in either of 
the given numbers. 

Also that any prime factor, as 0, which occurs in 
only one of the given numbers occurs the same num- 
ber of times as a factor in the product. 


(1) See Tannery, p. 134. 


GREATEST COMMON FACTOR. 203 


And that the number of times that any prime factor 
which is common to two or more of the given numbers, 
as a, occurs in the product is the sum of the numbers 
of times that it occurs in those numbers. $937: 

639. A convenient way of arranging the factors of 
the product given in the last article is to place together 
those that ar alike. We will then hav | 


VY. +++ = (4a --- (a, + a, + @,) times) 


x (88 --- b, times) (777 -- +(e, + ¢,) times) 
x (ddd... d, times) --- | 

640. Theorem. /f two positiv integers a and f be 
vesolud into prime factors, then, if a>> 8, every prime 
factor that occurs in B will also occur in aand the num- 
ber of times that it occurs in a will be not less than the 
number of times that it occurs in 8, and conversely, if 
every prime factor that occurs in 8 occurs alsoin aa 
number of times not less than the number of times that 
it occurs. in B, a >> B (’). 

To prove the direct theorem, we hav a = fy. 

The first part of the theorem is now evident from 
the first part of § 638. 

Next let 0 be the number of times that any given 
prime factor occurs in 7. 

If this factor does not also occur in 7, it will occur 
in the product fy, or in a, a number of times a, which 
is equal to J, by the second part of § 638. 

If this factor occurs also in 7, say ¢ times, it will 


(1) See Legendre, p. 6; Stieltjes, p. 12. 


204 INTEGERS. 


occur in a a number of times @, which is equal to 
6+ c, by the third part of § 638. 
In this case a> 4, § 75, 
Hence in both cases a = 8, 
Therfor at d, § 120. 
For the converse theorem, we may arrange the 
factors of a so that those which ar common to a and 
f shall come at the beginning, each the same number 
of times that it occurs in f. § 381. 
We then hav a = f x other factors. 
Therfor Lo> oe, 


For example let \"a — d0stcg 


and B= Off. 
We may then write a = (0£) (def) = B(deC). 
Hence “a> p. 


641. Theorem. /f two positiv integers a and f be 
resolud into prime factors, then, if a >> f, the quotient 
a: [, resolud into prime factors, will hav every prime 
factor that occurs in a but not in 8 the same number of 
times that it occurs in a, every prime factor that occurs 
the same number of times in a and B will not appear in 
the quotient ; and every prime factor that occurs in both 
aand 2 but more times in a than in B will occur in the 
guotient a number of times equal to the difference of 
these numbers of times. 

We hav a= fy anda: B =7. 

Since the prime factors of the product fy, or a, ar 
obtaind by putting together the prime factors of £ 


GREATEST COMMON FACTOR. 205 


and y, all the factors that a has but P has not must 
be factors of the quotient 7. 

Hence the first part of the theorem is evident. 

The second part of the theorem is true, for, if any 
prime factor occurd in both # andj, it would occur 
in the product a4 a number of times equal to the sum 
of these numbers (§ 638), that is, the number of times 
it occurd in a would be greater than the number of 
times it occurd in f. So75. 

To prove the last part of the theorem, any factor 
that occurs in both a and f, but more times in « than 
in must also occur in y. Otherwise it would occur 


the same number of times in a as in f. § 638. 
If, then, we let the numbers of these times be 
a, 6, ¢ respectivly, a=b+e, SHE ese 
Therfor Ces Denmn 
Thus, if a = d(eee..-e times) (Kxcx ..-#, times) 
and B = (eee ---e times) (Kee -.- 2, times), 


a> PB anda: 2 = d(«KcK---(k, — &,) times). 

642. We will now study the theory of the greatest 
common factors of series of two or more integers and 
will find the following theorem useful. 

Theorem. Zhe greatest common factor of two or more 
integers, of which at least one 1s not zero, 1s the same as 
the greatest common factor of those of the given integers 
which ar not zero, 

For every common factor of the integers which ar 
not zero is also a factor of the zero integers. 

643. In virtue of this theorem, when treating of the 


206 INTEGERS. 


greatest common factor of a series of two or more 
integers, we will suppose that none of them is zero. 
644, Theorem. Jf a fpositiv integer K 1s a common 
Jactor of a series of integers a,, A, Ug, ++. and every 
positiv common factor of G,, G,, O, +++ 1S a factor of kK, 
then « ts the greatest common factor of G,, Oy As, ++: 
Let 4 be any positiv common factor of @,, @,, @, ++: 
Then by the hypothesis « = gA and 4=« : g, wher 
¢ is positiv. 
INOW, ah ORK Oe ROT Ace ake § 493. 
That is, of the positiv common factors the one for 
which g = 1, which is x, is larger than any other. 
Therfor « is the greatest common factor of 
CNA aee 
645. Theorem. Having given any series of integers 
G,, Ay Gy, +++, 4, none of which is zero, uf we find the 
greatest common factor x, of a, and a,, then the greatest 
common factor K, of K, and A, then the greatest common 
factor Kk, of K, and @,, and so on, and finally the greatest 
common factor k__, Of. «._, and G,, then Kk _, ts the 


1 
greatest common factor of a, Gy, As, +++, &,,('). 
First, to show that «,, is a common factor of 
TOL, poms etl 


By hypothesis «, is a common factor of 4, and @,. 
Hence, since «, is a factor of «,, 

it is a common factor of a, and @,. § 460. 
And by hypothesis «, is a factor of a,. 


(2) See .Chrystal, p. 45; Tannery, p. 1153 Le Besgue, p. 32; 
Lucas, p. 345. 


+? 


GREATEST COMMON FACTOR. 207 


Hence «, is a common factor of 4, @,, 4. 
Since «, is a factor of «,, 


it is a common factor of a@,, 4, a. 


And by hypothesis «, is a factor of a,. 


Hence «, is a common factor of @,, 4, a5, @,. 


Similarly «,_, is a common factor of 4,, @,, @, +++, 


ea 


Second, every positiv common factor of 4,, a, 4 


29 73) 
eps isyaractor Ol vce, 
n—1 


For, let 2 be any positiv common factor. 
Then, since 2 is a common factor of a, and 4,, 
A is a factor of «,. § 602. 
Since A is a common factor of «, and a,, 
A is a factor of «,. 
Similarly we can prove that 4 is a factor of &,, 


ae) Ky 


Since, then, «,_, is a common factor of 4,, a,, a 


ar 


a, and every positiv common factor of 4, 4, G,, --- 


d 


a, is a factor of k,_,, «,_, is the greatest common fac- 


m1? <n 


COUR OLE 5, hoch, Clas § 644. 
This theorem might be stated : 
Having given any sertes of integers d,, Ay, Us +++, &,, 


none of which is zero, their greatest common factor ts 
(2,4) Q4)@: Qa. 

646. Theorem. //a, 8, 7 ar three integers, none of 
which is zero, (A@QP)@Wr=4W(FR7). 

By § 645 the first of these expressions is the great- 
est common factor of a, , and ;. 

So also is the second. 


For 2@(8W7) = (F@7r) W4 S574. 


208 INTEGERS. 


By §645 (?W&yr)@&a@ is the greatest common fac- 
tor ols -eandia: 

But this is evidently the same as the greatest com- 
mon factor of a, 8, and 7. 

Therfor (4@B) @r = «QBQ@?). 

This theorem is the Associativ Law for the Operation 
of finding Greatest Common Factor. 

The proof given above depends on the commutativ 
law. But it could easily be proved independently 
that a@(?&7) is the greatest common factor of a, 
8, and 7. Then the associativ law would be indepen- 
dent of the commutativ law. 

647. As a consequence of these two laws we can 
easily prove the following theorem, which is analogous 
to the generalized associativ and commutativ law for 
multiplication (§ 381), and which may be calld the 
Generalized Associativ and Commutativ Law for the 
Operation of Finding Greatest Common Factor. 

Theorem. Jz any complex expression in which inte- 
gers, none of which 1s zero, ar connected by the sign for 
greatest common factor, with parentheses to indicate the 
order in which the various operations ar performd, the 
arrangement of the parentheses and the order of the ele- 
ments may be changed in any way, the result of the 
whole set of operations being the greatest common factor 
of the series of integers. 

648. Hence we may write the expression 


aWP @W,We@:®&-::- without parentheses, and 


may write the elements a, f, 7, 0, ¢, +». in any order 


GREATEST "COMMON FACTOR, 209 


we please, the expression still denoting the greatest 
‘common factor of a, 8, 7, 0, €, --- 

SiO mea REOLOM 0 t/q andy == 6 =e) Ae, 
thenaWMrWeW7&::-=PCWIClWIW::: 

This may be proved in the same way as § 384. 

650. As a particular consequence of §§ 647, 649 it 
follows that in finding the greatest common factor of 
a series of integers, none of which is zero, we may re- 
place any number of them by their greatest common 
factor and find the greatest common factor of this 
greatest common factor and the remaining integers ('). 

651. Theorem. Jf none of the integers G,, ,, G., +++, 
a,, @ ws zero, |0|(a,€a4,@a,&---&4,) 

= (80) © (G4,) © (G45) & --- © (G4,). 

The proof of this theorem is formally the same as 
that of § 360. 

This theorem is the Generalized Left-handed Distri- 
butiv Law for Multiplication and the Operation of 
Finding Greatest Common Factor. 
| 652. Theorem. // none of the integers G,, Oy, Os, +++, 

a, 0 is 2070, (0, Q% Qa, Q---Qa,) || 

= (149) @ (a9) @ (UP) @ ++ Q (ah) 

This theorem is the Generalized Right-handed Dis- 
tributiv Law for Multiplication and the Operation of 
Finding Greatest Common Factor. 

653. Theorem. Jf the greatest common factor of a 
sertes of integers, none of which 1s zero, 1s unity, and 
these integers ar all multiphed by a given integer aif- 





(1) See Tannery, p. 115. 
tg 


210 INTEGERS. 


Serent from zero, the greatest common factor of the prod- 
ucts 1s the numerical value of this multiplier ; and con- 
versely, uf the greatest common factor of the products ts 
the numerical value of the multiplier, the greatest com- 
mon factor of the given integers ts unity. 

The proof of this theorem is like that of § 601. 

654. Theorem. /f mone of the integers 
LEURREIMGT Tiny Osa eben ely. nh cm Aa H), 
(4,€4,64,6 ae © 4,) (2,62, W8; & <s. & 3.) 
= (47;) & (4,81) & (4,,) & + © (4,2) 
QD (482) © (422) S (4592) @ +++ © (nis) 
© (4) @ (443s) © (44s) a ie (On ss) 


@ (u,) @ (4,/3,) @ (a8) Q- @ (ah 0. 
The proof of this theorem is like that of § 371. 
This theorem may be stated: The product of the re- 

Spectiv greatest common factors of two series of integers, 
of which none ts zero, 1s equal to the greatest common 
Jactor of all the products that can be formd by multi- 
plying an integer of the first series by an integer of the 
second, 

655. Theorem. Zhe product of the respectiv greatest 
common factors of any number of series of integers, of 
which none ts zero, 1s equal to the greatest common factor 
of all the products that can be formd, each having as a 
factor one and only one integer from each of the given 
Series, 





(1) See Stieltjes, p. 3. 


GREATEST. COMMON FACTOR. 7h | 


The proof of this theorem is formally the same as 
that of § 380. 

656. Theorem. // none of the integers a,, O,, As, 
ws zero and they ar all divisible by an integer 0, then 
a,(© 4,6 4,&--- >> 4 and (4, @Qe,64,@---):|F| 
= (1%: VQ (ty? )@ (a, )Q---C) | 

This theorem is aeaneal from § ae just as § 602 1s 
proved from § 599. 

This theorem is the Generalized Distributiv Law 
for Division and the Operation of Finding Greatest 
Common Factor. 

657. Theorem. // al/ of a series of integers, none of 
which 1s zero, ar divided by their greatest common fac- 
tor, the greatest common factor of the quoticnts will be 
_ unity ; and conversely, tf, when all of a series of integers, 
none of which ts zero, ar divided by a given integer, the 
greatest common factor of the quotients is unity, the 
numerical value of the given integer is the greatest com- 
mon factor of the given series of integers. 

The proof of this theorem is like that of § 603. 

658. This theorem is equivalent to the following : 

Theorem. Having given a series of integers O,, Ay, Fs, 

.-, of none of these integers 1s zero and K 1s their greatest 
common factor, ther exists a sertes of Integers 71, Foy T a °° * 
none of which 1s zero and whose greatest common factor 
PTL ASU CICIIUL Ceeaay Keep eater a C700, 
COMUG SOY, 7) Ci 70K, 0, 7K l= 7K, 1, WHEY HONE 
of the integers Kk, 7, To Ty +++ tS Zero and the greatest 


(1) See Stieltjes, p. 2. 


z2Ii2 INTEGERS. 


common factor Of 147s) 7 +++ unity, then « ts the greatest 
common factor of O,, Gy, Gs, ++ 

659. Theorem. Zhe common factors of a series of 
integers, none of which is zero, ar the same as the fac- 
tors of their greatest common factor ('). 

660. Theorem. // two of the integers a,, G,, O,°-+, 
none of which ts zero, ar prime to each other, the great- 
est common factor of the series ts unity. 

To find the greatest common factor of the series of 
integers, we can arrange them as we please. § 647. 

Call, then, the two that ar prime to each other a, 
and 4,. 

Now, since 4, || @,, 4,Q)a, = I. 

Then 4, © 4,6 4@Q ++: = (4&6 %) & (4, @ - ++) 

§ 647. 
! ie (4,@Q-++)  § 583. 
§ 577. 

661. Thepren Lf two of ‘. INLEZEVS Op Ay, O,; 

none of which is zero, ar prime to each Sian oon G zs 


any integer except zero, then 
(4,9) & (4,4) & (2,0) & es are | Y . 

This follows immediately from the preceding the- 
orem and § 653. 

662. Definition. A series of two or more integers 
ar prime to each other two by two(*), when each of 
these integers is prime to each of the others. Thus 
2, 5, 9 ar prime to each other two by two. 





(?) See Tannery,’ p. 112. 
(2) See Chrystal, p. 38; Tannery, p. 117. 


+e 


GREATEST COMMON FACTOR. 213 


When two integers ar prime to each other, their 
greatest common factor is unity, and the converse is 
true (§ 579). 

When three or more integers ar prime to each 
other two by two, their greatest common factor also 
is unity (§ 660), but the converse is not always true. 

Thus the greatest common factor of 3, 5, 9 is 
unity, but these integers ar not prime to each other 
two by two, two pairs, 3 and 5, 5 and 9, being prime to 
each other, but the third pair 3 andg not being prime to 
each other. 

Again the greatest common factor of 6, ", d is 
unity, but no two of these integers ar prime to each 
other. 

The statement that ‘‘ When three or more integers 
ar prime to each other two by two, their greatest 
common factor is unity”’ is not, then, a complete an- 
alog of § 570. 

We will, however, deduce in § 666 a theorem which 
is a complete analog of § 579, a theorem which is 
true of a series of integers and which includes that of 
§ 579 as a particular case. 

663. Theorem. /f v >> Ga,, v > Ga, v>> Oa, --., 
y>> Oa, and the integers a, Ay, G,, +++, a, ar prime to 
each other two by two, then v >> baa,a,.-.a, and 
|» | : | Pa,a0, +++ 4, |= : da) 6 ‘ Oa,) © : Ga,) 
& as ‘Wl 2 Oa,), provided v is not zero; and con- 
versely, if v>>Oaa,a,--.a,, then v>> ba, v > ba, 


n? 


v>> a, --, v>Oa, and, f |v|:|Oa,a,a,---a, | 


214 INTEGERS. 


= (¥ : by) @(v + 6a) @(v 04) @++ Ql : 6a), 
then the integers G,, Gy, Os, +++, a, ar prime to cach other 
two by two. 

For the direct theorem, 
SINCE) BL> 0G) tye 0) ean SG meee eu as 
and Vy Pai, = (y 00) OO (UG ye zoe 

That is, the theorem holds for two integers. 

To prove it in general, suppose that it holds for & 
integers, wher 2 =L< 2. 


That is, we hav given vy >> @a,a,a,-..a, and 


|» | . | 44,4,4,---4,| = (v : J4,) QU : Jay) QL : Gas) 
OO.) (vis Ga,). 
Then “@ia.q7-s% a \iiaeer. § 632. 
Hence v >> 0(4,4,4, +++ a,)a,,, and 
| 4 | : | O(a, 00, arog A, )Oy 4 | 
(0 Ga ae ee) Ova) § 628. 
= (|»| ; | Pa,4,05--+ 4, |) QO Uaes) § 580. 
= (Aa) Qv : Oa) Qe 0a, Q--- 
Wy: Ga,) Ql : Ga,,:).  § 583. 


Hence, if the theorem holds for & integers, it holds 
also for &+ 
Therfor by the law of induction it holds for z integers. 


For the converse theorem we hav given 
OR DES OE Roo io The 


Hence v >> da,, » >> Oa,, v >> Oa,, --., v >> Oa,. 


§ 460. 


Hence also »v = Afa,a,a,-..a,, wher A + 0. 


Therfor v : @a,a,a,---a, =A 


GREATEST COMMON FACTOR. 215 


and PRE eed lps a. a 
ys Ga RO, +++ 0. 
y: Ga, = aa, ++-a a 
v. Ga, are 20,00, caste Guy 
Hence 


y:da)W(: 6a, Wr: Ja,)®---&(: 6a,) 
am (Aaa, + +++, 14,,) & (Aaa, +++ a, 44, 4 
& (Aa, 4, st ee ,,) &:: ‘& (Aaa, pct oo a4) § 649. 
— | A | [(4:43-+- a,_%,) & (4,4, ++ ate 
& (4,4, PME Ps Z,,) &-- ‘& (4,4,4 ee a, 4) ]. § Oot. 


But we hav given also 


|» | : | 0a,a,0,+-+6 





n 


poy Co canes) CO See) Ce -‘&(v: Oa). 
Hence | A | a | A | [ (44 We Fo of) & (2,4; - a, \4,) 


& (4,4, DLE cay ay &-- ‘& (2,24, ° ie amy : 
Therfor 


(4,4, eee eat is) & { (4,4, - ra a,_,2,) ) (4,4, wae CLS cea Z,,) 
&-- © (44, d+ +- a, 1)} =a ale 
Or 


(4,0, a ont a,,) & { 4, | [(4; Cie 1%n) & (a, ie n) 
Co .@ (G45 - i may aes 
thats. 


NS eR AEA | a, IL | 4, IL(4; - el 4.) © (4: PEE os a,) 


&:: ithe < en Ob § 579. 
iherior ag) a0 Fala. §§ 631, 561. 


Therfor a, is prime to each of the integers @,, a,, ++», 
CH a, § 63 2. 


216 INTEGERS. ‘ 


Similarly it may be shown that each of the integers 


G.,, Us, +++, @, 1S prime to each of the other integers. 
That is, 4,, 4, @,, +++, @, ar prime to each other two 
by two. § 662. 
664, Theorem, 79) Ss ae eee 
and the integers G,, G, Gs, ++. ar prime to each 
other two, by = two, \ ther Vo Oa eased, 


y 








| ait, |= 0:4 OC Q)OC: YQ“, 
provided v 1s not zero; and conversely, tf 
VES OO, 0ac: NCH OY A ee eed ae 

JY] | %25--- |= 14 )OYC:4)WU:i4)@--, 
then the integers G,, G,, O,--. ar prime to each other 
two by two('). 

This theorem is a particular case of the preceding, 
the case when @ = 1. 

665. Theorem. /f v= 6a,a,a,:-- and ithe 
inlegeYS Ay, Ay Os,+-+ ar prime to each other two by 
two, then v= 0d,, Yi Ua, 0a re 
(v : 0a,)\@®(¥: 04,)&Q(v : 6a,)& +--+ = 1, provided none 
of the integers 0, a, A, G++. 1S zero; and conversely, 
yf) (¥3 Pa) OOM 20a) COW. Ua.) — 1 eee 
inLEgEVS Oy, Ay, Os,+++ ar prime to each other two by two. 

This follows immediately from § 663. 

FOr sil Sr 000 Gee een ey > 00 eee 


ie oe 


yi Ua,,---, Vise 0a.c,¢,~-- wand? yi) 0G a a aele 
666. Theorem. /f »=a,a,a,:-- and the integers 
4, Gy Gs,-++ ar prime to each other two by two, 


then (9250 OS) (sy Ge) (COs a OO terre Be 


(1) See Tannery, p. 120, 


GREATEST COMMON FACTOR. 217 


provided none of the integers a, 4, @ 


Fe igi an area 
ws zero; and conversely, ff v= a,a,a,--. . and 
Y24)O:4,)O:4,)Q---=1, the imtegers 
Ay, A, As, --- ar prime to each other two by two. 


This theorem might also be stated as follows: 

Lf the integers a, a, Gj, +++, 4,, none of which ts 
zero, ar prime to each other two by two, the greatest 
common factor of the n products that can be formd, each 
containing all but one of these integers, is unity; and 
conversely, tf the greatest common factor of these n 
products 1s unity, the given integers ar prime to each 
other two by two. 

For if Y¥=2,0,0, «= 


Y3>0,=4,0,°°-, 


EN A,A,++° 


b 


YL, = A4,°°:, . 


This theorem is the analog of § 579 mentiond in 
§ 662. 

It contains § 579 as a particular case. 

Ope P ra Oya ean Yet a, 

Thus we hav from this theorem that, if «@,||@,, 
a, @&) a, = 1, and conversely. 

The reason is now apparent why, in order that a 
series of z integers, when z > 2, may be prime to 
each other two by two, it is not sufficient that their 
greatest common factor shall be unity. The greatest 
common factor of the z products, each containing all 
but one of the integers, as above, must be unity. 

But when z= 2, the z products ar the integers 
themselves, so that the two conditions ar the same. 


667. Theorem. Jf « zs the greatest common factor of 


218 INTEGERS. 


the integers a, Ay G,+++, a,, mone of which ts zero, a 
series of integers kh, Ay hy +++ 4, can be found, such 
that 


Ai, + Aga, + Ag%g +++ +A, = K (’). 
For, let «, be the greatest common factor of a, and 
nek BOR MalieeemcLt: 
Then Kk, =a, + 6,4, 
Ky = YK, Sols, 
K, = YK, + Sieh 
(Pit pelt iis og ak 51% y 
. § 610. 
From the first two of these equalities 
sg V(r, 4, ats Fy) aa 0 Nite V5 4, ar F405. 
Combining this with the third, we get 
K, = Vd O, + VVE A, te VE ot, 3F F 0L,. 
By examining this formula, we see that it is a 
particular case of the formula 
ey YY Ved, PY 1+ Y3YS Fg 
eee VE 9g aeons VY, 8 1 ar ve, 1% 7 5 1Fy41) 
thercaseswhens ==-934 
Similarly the formulas for «, and «, ar particular 
cases of the latter formula. 
We will now show that this formula holds for all 
values of / from I up to z— 1. 
Suppose that it holds for some particular value /, 
whet © =a) =47 115 





(1) See Gauss, p. 33. 


Lal 


GREATEST COMMON FACTOR. 219 


nen Since Kievan Kats 0, we hav 


Keg ea YY FY Yi VVF Oy 


dt hat VE oly nF AT Sg De eM Aim 
aE Daas ar Bis Ch ts oe 

But this is the same as the formula for «, with 7+ 1 
put in place of 7. 

If, then, the formula for «, holds for any particular 
value of Z, it holds for the next. 

‘Hence, since we hav shown that it holds for the 
values I, 2, 3, by the law of mathematical induction 
it holds for all values of 7 from I up to the last value 
n—I. 

We hav, then, finally 

ile eM es Sigmon gt fet ot VeVF 
ar eeeebe ery VE 0s a Pi tear: Doth aah) > 
ap er esi en, 


But eee 1c, 


Hence, setting A=) 2 ey 
A, poy ae) peat Vis, 
, gee ar pe © Does 


Pa care ee ; 
we hav «=A a,+/,0,+ 4,0, +---+4,4,. 

668. Theorem. /f wmnity 1s the greatest common 
factor of a series of integers O,, Ay Gs, +++, &,, none of 
which 1s zero, another series of integers h,, Ay Ay -++, A, 
can be found, such that 


1a, + Aja, + A,a, + +++ +A a, = I. 


Paes, @) INTEGERS. 


669. Theorem. /f a,+4,+4,+---=1, then 

This is proved in the same way as § 612. 

670. Theorem. /f o,+4,+4,+---=1 and fi, 
Box Bes 20: OF fACLOVS Of GO, danas HESPECHUL) aleve 
2. C2WR &: Pena k 

671. Theorem. /f ja, + Aa, + 4,2, +++-= 1, the 
greatest common factor of any series of integers taken 
one from each of the groups 1, and a,, i, and a, i, and 


gener 2s unity. 
672. Theorem. 
If ha, + dg, +ia,++--=4,@€4,€a,®:-- 


then 4, OO A, 004209 21. 


This is proved in the same way as § 6160. 

This theorem may also be stated: Jz whatever may 
the greatest common factor of U,, A, Os, +++ 1S exprest in 
the form ha, + 4,0, + 40, + +++, the greatest common 
factor of hy hy 4g +++ ts unity. 

673. Theorem. Jf « zs the positiv integer which has 
as prime factors all the prime factors that ar common to 
a series of positiv integers, and which contains each 
prime factor a number of times equal to the least num- 
ber of times that it occurs as a factor in the given in- 
tegers, then x is the greatest common factor of those 
mntegers. 

First, since each of the given integers contains every 
prime factor that occurs in « a number of times not 
less than the number of times that it occurs in x, x is 
a common factor of the given integers. § 640. 


+f 


GREATEST COMMON FACTOR. 221 


Now let 4 be any positiv common factor of the given 
integers. 

Then every prime factor of 4 must be common to 
the given integers and must occur in 4 a number of 
times not greater than the number of times that it oc- 
curs in any one of the given integers ($640), and 
hence not greater than the least number of times that 
it occurs as a factor in the given integers. 

Therfor every prime factor of 4 occurs in « a num- 
ber of times not less than the number of times that it 
occurs in A, 

Hence 4 is a factor of x. 

Since, therfor, « is a common factor of the given 
integers and every positiv common factor is a factor of 
k, « is the greatest common factor of the given integers. 


[Ciebaes lisse 11, 


LEAST COMMON MULTIPLE, 


674, Definition. A common multiple (") of two, or 
more, integers is an integer that is a multiple of each 
of them. 

Thus zero is a common multiple of any two, or 
more, integers, none of which is zero ; the product of 
two, or more, integers, none of which is zero, is a 
multiple of each of them; also their product multi- 
plied by any factor. 

675. Let a, 8, 7, --- be any two, or more, integers, 
neither of which is zero, and 4 any integer not zero. 
Then dafy--- is a common multiple of a, f, 7, -:: 
Now of all the values of Aafy --- obtaind by giving 4 
various values the numerically smallest ar those for 
which |A| = 1. For, 
ieee vee oT, | AaBy ---| > jaBy--- |. § 406. 

Ther may be common multiples of a, 2, 7, --- which 
ar numerically greater than af; ---, but which ar not 
multiples of afy.-. Thus 9 x 4, or 6 x 6, which is 
greater than 4 x 6, is a common multiple of 4 and 6, 
but is not a multiple of 4 x 6. 

Moreover ther may be common multiples of a, 8, 


(1) See Stieltjes, p. 2; Tannery, p. 121. 
222 


LEAST COMMON MULTIPLE. 223 


y, ---numerically less than the product a4fy--- Thus 
3 x 4, or 2 x 6, which is acommon multiple of 4 and 
6, is less than the product 4 x 6. 

676. The number of positiv integers not greater 


than |afy---| is |afy---;. Hence the number of 
positiv common multiples of a, 9, 7, --- not greater 
than |af7---| is not greater than |afy---|. It is, 


therfor, limited. Of these multiples any one is less 
than any positiv common multiple greater than 
| a7 ---|. The smallest of them is therfor the small- 
est of all the positiv common multiples of a, 8, 7, --- 
It is calld the least common multiple of a, f, 7, --: 

Definition. The least common multiple of two, or 
more, integers, none of which is zero, is the smallest 
of the positiv common multiples of those integers. 

677. If we had defined the least common multiple 
of a set of integers as the numerical value of the 
numerically smallest of their multiples, the least com- 
mon multiple of any set of integers would be zero. 

The idea of least common multiple would not, then, 
be anything new. The idea as defined above, how- 
ever, is a new and useful idea. The least common 
multiples of different sets of integers ar not always the 
same, as may be seen by taking a few examples. 

It is to be noted also that the least common multi- 
ple is not defined except for a set of integers, none of 
which is zero. This is in accordance with the restric- 
tion of § 427. 

678. Theorem. Jf a positiv integer is a common 


224 INTEGERS. 


multiple of a series of integers O,, Oj, Ay, ++» and every 
positiv common multiple of 0, A, A, ++. 1s a multiple of 
pt, then pis the least common multiple of 0, 4,, A, ++: 

Let » be any positiv common multiple of a,, @,, a,, ++ 

Then by the hypothesis y = ep, wher ¢ is positiv. 

Now, if'g-> 1,0 (eve OL Ven 

That is, of the positiv common multiples the one 
for which g = 1, which is yp, is less than any other. 

Therfor # is the least common multiple of a,, a,, 
Toes 

679. We will denote the least common multiple of 
two integers a and # by the symbol a X f, which may 
be read “a mul f.” The symbol a X # will, then, 
be univalent, if neither a nor f is zero. The symbols 
aX 0,0 X a,ando X o will, however, be meaningless. 

The integers ¢ and # may be calld the elements of 
the least common multiple a x f. 

680. Operation of Finding Least Common Multiple. 
The least common multiple of two integers, neither 
of which is zero, is found by choosing the smallest 
among a given limited set of integers. It is therfor 
found by an operation. 

681. Theorem. a X B=[ Xa, provided neither a 
nor B 1s sero. 

The proof is like that of § 574. 

This theorem is the Commutativ Law for the Opera. 
tion of Finding Least Common Multiple. 

682. Theorem. /f a>, axXP=| al, frowded 
neither a nor 3 ts zero, 


+? 


LEAST COMMON MULTIPLE. 225 


The proof is like that of § 575. 

683. Theorem. « X «4=|«a|, provided «a 1s not zero. 

684, Theorem. « X I = |a|, provided a is not zero. 

685. Theorem. 

(<4) x B=ax (—f) =(-4) x (—A)=4XB 
provided neither a nor 8 ts zero. 

686. Theorem. /f a= 8,aXy=§ X 7, provided 
neither [2 nor x ts Zero. 

687. Theorem. /f a=, y Xa=7 Xf, provided 
neither 3 nor 7 1S Zero. 

GSsen theorem md — oad * 0, a X18 xX 0; 
provided neither 8 nor 7 ts Zero. 

689. Asthe least common multiple of two integers 
is an integer, we may combine it with other integers 
by any of the signs +, —, x, :, &, X; provided, 
first, that the first element of every quotient is divis- 
ible by the second ; second, that of two integers con- 
nected by the sign ( not both ar zero; and third, that 
of two integers connected by the sign  X neither is 
_zero. We will use parentheses to indicate the order 
in which the various operations ar performd. 

Every such expression is univalent. For every 
sum, difference, product, quotient, greatest common 
factor, and least common multiple is univalent. 

690. Theorem. Jf for one or more elements of a com- 
plex expression which contains integers connected by any 
or all of the signs +, —, x, :,®, X and in which, 
jirst, the first element of every quotient ts divisible by 
the second element, second, of two integers connected by 

1S 


226 INTEGERS. 


the sign ( not both ar zero, and third, of two integers 
connected by the sign X nether 1s zero, equal integers 
ar substituted, the complex expression 1s unchanged. 

691. Theorem. « X S=|af|:(4Q&), provided 
neither a nor 3 ts zero(*). 

Since neither @ nor 3 is zero, | a | is positiv. 

Hence | af |: (4 @ ) is positiv. 

Moreover !48|:(4@) =|4|(|8|: (4@B)) § 468. 

=(\a|: @QAAl: § 467. 

Therfor | a8 | : (2@) is a common multiple of | «| 
and | #|, and hence of a and f. 

Moreover every common multiple of a and f isa 


multiple of | 4|: (4@ ). § 630. 
Therfor | a8 |: (4 @ #) is the least common multiple 
of a and f. § 678. 


By means of this theorem the least common mul- 
tiple of two integers, neither of which is zero, may be 
easily found when their product and greatest common 
factor ar given (°). 

692. Theorem. Axy common multiple v of two in- 
tegers a and 8 ts a multiple of thew least common 


multiple, 
If »y is zero, the theorem follows from § 444. 
If vis not zero, vy >> |aP|: (a@ 6). § 630. 
And |48|:(@«&@f)=a4 xB. § 691. 
Hence > a 7. 


(1) See Le Besgue, p. 34; Tannery, p. 122. 
(2) See Stieltjes, p. 4. ) 


LEAST COMMON MULTIPLE. 227 


693. Theorem. Zhe common multiples of two integers 
ar the same as the multiples of their least common mul- 
tiple ('). 

For, by § 460, every multiple of the least common 
multiple is a multiple of each of the given integers. 

And, by §692, every common multiple of the 
given integers is a multiple of their least common 
multiple. 

Hence the two groups of multiples, first the com- 
mon multiples of the two integers and second the 
multiples of their least common multiple ar the same. 

694, Theorem. (a X P\aWPs)=|a8|, provided 
neither a nor 3 ts Zero. 

This follows easily from § 691. 


As a consequence of this theorem ther is a reci- 
procity or duality (°) between the theories of greatest 
common factor and least common multiple. It will 
be found that many theorems in these subjects ar still 
true when the terms “greatest common factor” and 
“least common multiple” ar interchanged. 

695. Theorem. a2@f=|af|: (a x §), provided 
neither a nor 8 ts zero. 

By means of this theorem the greatest common 
factor of two integers, neither of which is zero, may 





(1) See Tannery, p. 122. 

(2) See L. Cremona, ‘‘Eléments de Géométrie Projective,’’ 1872, 
English translation, ‘‘ Elements of Projective Geometry,’’ 1885, by C. 
Leudesdorf, p. 26; Beman and Smith, ‘‘ New Plane and Solid Geom- 
CY O01... 27- 


228 INTEGERS. 


be easily found when their product and least common 
multiple ar given. 
696. Theorem. 
If v is a common multiple of a and 3, then 
[eo 4) X %: A) (0:4) @e: AH) =|: 148, 
provided v ts not zero. 
For [(v: 4) X@:A)] [2:9@QC:f)] 
=|(¥: 4): B)| § 604. 
=| (vv) : (a) | § 466. 


697, Theorem. 
Tf v ts a common multiple of a and 3, then 
(2X Pv: 246: P)] =|», provided v ts not zero. 

By § 630, |»|: (a2 |: (@@QPA) =: 9 Qe: A) 

But | ap 63 (@ C02) == ox eso 

Therfor | | (a. 8) = (hs 2) OO Care): 

Or (4X AY: 4)@e:A)] =|¥1, 

This theorem includes § 694 as a particular case. 

HOt, Sift 10 0 ee ee 

698. Theorem. Jf » zs the least common multiple of 
the integers a and , then (v:4)@Q(v: Bf) =1; and 
conversely, uf the posit integer v 1s a common multiple 
of the integers «and 8 and (v:a)®(v: 8) = 1, then v 
is the least common multiple of a and fi. 

Both parts of the theorem follow easily from § 697. 

699. Theorem. Jf » 2s a common multiple of a and 
B, then (2@B)[%:4) Xe: P)] =|»|, provided v is 


not Zero, 


=| vy]: jaB |. § 496. 


+ 


LEAST COMMON MULTIPLE. 229 
For 


(4X A{ZWAlY:4 XP: ALC: 9@QL:A] 

= {(4 X B)(4@8)} 
x{[%:4) XP: A][Y:4 QC: A)]} $38. 

eek (ee ts ao |) =| ye! $$ 694, 696. 


Moreover 


(4X P{Z@ALY: 4) xv: AIC: JQ: /)] 
= ((@@A[e:2) xb: A)])} 
A IOC Ge ttaaes § 381. 
ee Co) sa) xX (y 2B) } ||. § 697. 
Hence Weg Ne Qa) OR) ley) == py be 
Therfor — (4@A)[o: 4) x @18)] =|». 
700. Theorem. /fal|| 8, aX 8=|af|, provided 
neither a nor [3 ts zero, and conversely. 
This follows easily from §§ 5709, 601. 
701. Theorem. (4) X (a7) =|a| (2 X 7), provided 
neither a, 8, nor 7 18 Zero. 
For (48) x (a7) = | (a) (ar) | : (af) @ (a7) § 601. 
=(]4| | | Kee (|| (8 @7) 
= (|4||8r|):@@r) § 457. 
=|4|(4r1: @@n)=l4|/4 x7). $8 468, oor. 
This theorem is the Left-handed Distributiv Law for 
Multiplication and the Operation of Finding Least Com- 
mon Multiple. 
702. Theorem. (a7) X (87) = (a X f)|7|, provided 
neither a, 8, nor 7 ts zero. 
This theorem is the Right-handed Distributiv Law 
for Multiplication and the Operation of Finding Least 
Common Iultiple. 


230 INTEGERS. 


708. Theorem. /f 8 || 7, (48) X (a7) =| @f7 |, pro- 
vided neither a, 2, nor 7 ts Zero. | 

For (a8) x (a7) = |@|(2 X 7) § 701. 

=| 4| | Br | =| 48r'|. § 700. 

704. Theorem. /farsy and B>y7, thnaX BOY 
and (4X P):|\r|=(a:7) X (8:7), provided neither 
a nor 2 ts Zero. 

This theorem may be proved in the same way as 
§ 602, or it may be proved from § 602 by means of 
§ 691. 

This theorem is the Distributiv Law for Division and 
the Operation of Finding Least Common Multiple. 

705. Theorem. /f @ || 8, (a0) X B=(a X f)| 9], 
provided neither a, B, nor 0 is zero. 

This may be proved from § 625 by means of § 601. 

706. Theorem. Jf art-@ and O\|| 8, then 
(2:0)X B=(aX §#):|0|, provided neither a nor B 1s 
Zero. 
This may be proved from § 626 by means of § 6901. 
707. Theorem. faving given any series of integers 
Gz, As,+++, a, none of which ts zero, of we find the 


n) 


Th 
least common multiple yp, of a, and a,, then the least 
common multiple p, of p, and a,, then the least common 
multiple 1, of , and a,, and so on, and finally the least 
common multiple p,_, of p,_, and a, then p,_, ts the 


ine : 
least common multiple of 0, A, Gs, +++, a, (*). 


==1) 


First, to show that #,_, is a common multiple of 
a, Dy as, OO LL 


n 


(}) See Tannery, p..122 5 Lucas, p.7345. 


LEAST COMMON MULTIPLE: 231 


By hypothesis #, is a common multiple of a, 
and 4, 

Hence, since #4, is a multiple of y, 
it is a common multiple of a, and a,. § 460. 

And by hypothesis y, is a multiple of a,. 

Hence yz, is a common multiple of a,, 4,, @,. 

Since #4, is a multiple of 1, 
it is a common multiple of a,, 4,, 4,. 

And by hypothesis y, is a multiple of a,. 

Hence yz, is a common multiple of @,, @,, a, @,. 

Similarly ,_, is a common multiple of 4, a, 4,, 
ene 

Second, every positiv common multiple of a,, @,, a,, 
a, is a multiple of #_.. 

For, let vy be any positiv common multiple. 

Then, since » is a common multiple of a, and a,, 
vy is a multiple of #4. § 692. 

Since vy is a common multiple of y, and a,, 


oe 
5 


vy is a multiple of y,. 

Similarly we may prove that » is a multiple of 
(EN Ts Dele 

Since, then, #,_, is a common multiple of 4@,, 4,, @,, 

--, @ and every positiv common multiple of a,, 4,, 

Geo sista multiple of 7 jz is the least com- 
monemultiple of a.,-a,,0,, »°+, @.. | $ 678. 

This theorem might be stated : 

Flaving given any sertes of integers t,, A,, Ay +++, O,, 
none of which ts zero, ther least common multiple ts 


(4, xX ay) ras a) Pe ‘) X 4,. 


282 INTEGERS. 


It will be noticed that this theorem and its proof ar 
identically the same as those of § 645, except that the 
words “least”? and “multiple” replace ‘‘ greatest”’ 
and ‘‘factor’’ respectivly, the letter replaces «, and 
the symbol X replaces Q. 

708. Theorem. J/ a, f, y ar three integers, none of 
which 7s ero, Aja XB) X 7 = a AB OS 7). 

The proof is similar in form to that of § 646. 

This theorem is the Associativ Law for the Opera- 
tion of Finding Least Common Multiple. 

709. As a consequence of this theorem and that 
of § 681 we can prove the following theorem, which 
is analogous to that of § 647 and which may be calld 
the Generalized Associativ and Commutativ Law for the 
Operation of Finding Least Common Multiple. 

Theorem. /2z any complex expression tn which inte- 
gers, none of which 1s zero, ar connected by the sign for 
least common multiple, with parentheses to indicate the 
order in which the various operations ar performd, the 
arrangement of the parentheses and the order of the ele- 
ments may be changed in any way, the result of the 
whole set of operations being the least common multiple 
of the series of integers ('). 

710. Hence we may write the expression 
ax pBxrx 0 xe X=«= without parentheses; and 
may write the elements a, f, 7, 0, ¢,-+-in any order 
we please, the expression still denoting the least 
common multiple of a, f, 7, 0, ¢, -: 





(1) See Le Besgue, p. 34. 


+t 


LEAST COMMON MULTIPLE. 233 


(Pre neoretign)/ di f07 0, © = C1 7 = OF, 
ier ee KT XK = P COX CX Y--- 

This may be proved in the same way as § 649. 

712. As a particular consequence of §§ 709, 7II 
it follows that in finding the least common multiple 
of a series of integers, none of which is zero, we 
may replace any number of them by their least com- 
mon multiple and find the least common multiple 
of this least common multiple and ‘the remaining 
integers ('). 

713. Theorem. // none of the integers a,, 4,, Oy ---, 
a,, 0 ts Zero, 

10 | (a, X a X a, X ee) 
rae (Ga,) xX (da,) xX (Ga,) Ma Tee Pas (9a,,) (’). 

The proof of this theorem is formally the same as 
that of § 651. 

This theorem is the Generalized Left-handed Dis- 
tributiv Law for Multiplication and the Operation of 
Finding Least Common Multiple. 

714, Theorem. J// none of the integers a,, A, Ay ++, 
a,, 0 ts zero, 

(a, X a, X 4, X pera.) 2 | 
= (a,0) xX (a0) X'(a,0) X --- & (4,8). 

This theorem is the Generalized Right-handed Dis- 
tributiv Law for Multiplication and the Operation of 
Finding Least Common Multiple. 

715. Theorem. J// none of the integers a,, A, A+, 


Le ee toe ae NS) oe 1s ZEVO0, 


(1) See Tannery, p. 123. (2) See Tannery, p. 124. 





234 INTEGERS. 


(a, Xx Dig. Og X 4,,) (Pre tieX Bs Phe LO Pah B,.) 
oe (4,,) x (4,/3,) 78 (4,/3,) Paes Te aS (4,2) 
X (42) X (Ai) X (4532) X +++ X (432) 
. (4P5) xX (aaa), xX (4s) 2 Pak Gn °s) 


x (a4) X (08) X (0A) X +» X (Mah) 

The proof of this theorem is like that of § 654. 

This theorem may be stated: Zhe product of the 
respectiv least common multiples of two series of integers, 
of which none is zero, 1s egual to the least common 
multiple of all the products that can be formd by mul- 
tiplying an integer of the first series by an integer of 
the second. 

716. Theorem. Zhe product of the respectiv least 
common multiples of any number of series of integers, 
of which none ts sero, 1s equal to the least common mul- 
tiple of all the products that can be formd each having 
as a factor one and only one integer from each of the 
Liven Series. 

The proof of this theorem is formally the same as 
thatwotys 655: 

717. Theorem. /f none of the integers a,, d,, A, ++: 
7s zero and they ar all divisible by an integer 0, then 
GX a,X a, x .-- > 0 and 
(a, X a, X a X +++): |G 

= (4, : 9) X (4: 8) X (a: 8) X--- 

The proof of this theorem is formally the same as 


that of § 656. 
(1) See Stieltjes, p. 3. 





LEAST COMMON MULTIPLE. 235 


This theorem is the Generalized Distributiv Law for 
Division and the Operation of Finding Least Common 
Multiple. 

718. Theorem. Zhe common multiples of a series of 
integers Ay, A,, Ox, +++, &, ar the same as the multiples of 
their least common multiple (°*). | 

Othe errii ies, eee be the series) of least 
common multiples found as in § 707. 

Then the common multiples of a, and a, ar the 
same as the multiples of y,. § 693. 

The common multiples of 4, and a, ar the same as 
the multiples of /,. 

Hence the common multiples of a, 4, a, ar the 
same as the multiples of y,. 

Similarly the common multiples of 4, @,, a, @, ar 
the same as the multiples of w,. And so on. 

Finally the common multiples of @,, @,, a, +++, @, 
ar the same as the multiples of 4,_,, which is the least 
common multiple of @,, @,, @, +++, @,. 

719. Theorem. /f » zs a common multiple of the 
INLCLCYS Uy, yy Ag+++, A, then 
(%, X 4 X 4 X +++ X 4,) 

x [ev . A) Ov . a) OU ° a,)@Q--- WL ° a,,) | 

= |v|, provided v ts not zero(’). 

By $697 (a, X a) [b: 4) @(v: 4)] = [>], 


That is, the theorem holds for two integers. 





(eyoce te barricn, -arhes7s, Vol. Iil., 1383, p. 2173 Stieltjes, 


aes 
(2) See Barrieu, p. 217. 


236 INTEGERS. 


To prove it in general, suppose that it holds for % 
integers, wher 22k <n. 
Then we hav 


(a, X @, X a, X -+» X @,) 
x [VY 4)OY:4)OY: 4%) WB: We: &)] 


Sah 


Hence (v:4,)@Q(%: 4) WY: &)W-:- WY: 4) 
=|v|:(4X a Xa, X +--+ X 4). 
Hence 


¥:4)QY:4)OY:4) 6: WY: 4) 6 2 Way) 
=({¥|2(% Kil, Koy Ke a.) OO ea) 
ma (VG Kaa ae OO eee 
| » 2 (GK Gp Xray Kner) Cae) § 697. 
| 


Therfor 


(a, X a, X 4 X 3 KO, XG.) 
x[%:4)WY: 4)WY: a) @-:-- 
ae | | Ge oa) & (¥ = @,.4)] 


If, then, the theorem holds for % integers, it holds 
also for & + 1. 
Therfor by the law of induction it holds for xz 


integers. 
720. Theorem. lf v «aw the least common 
Multiple Of the) \ HLCP Crs A ane a. 


then (¥:4)\@QY:4)We:4)®Q-:- Wy cave oo 


LEAST COMMON MULTIPLE. 237 


and conversely, if the positiv integer v 1s a common 
multiple of the integers a, a, a and 


"p 2) 3) 


(¥:4)Q:4)@OU:4)@-:- WU: a then 


y is the least common multiple of a, U, G, +++, &,,('). 

Both parts of this theorem follow easily from the 
preceding theorem. 

721. This theorem is equivalent to the following, 
which is the analog of § 658. 

Theorem. Saving given a series of integers 
y ***) Gy, none of which ts zero; of pe is their 
least common multiple, ther exists a_ series of 
Integers 1 To Ty °°" Tn Mone of which is zero and 
whose greatest common factor is unity, such that 


CA 0 


eee eA) ) CONUEVSELY, 
Se eh Pe UEY » HONE OF 
the integers ¥1) To Ty **T, 2% sero and thew greatest 
common factor ts unity, then pw is the least common 
multiple of O,, Ay, As, +++, G,. 

722. Theorem. Having given a series of n integers, 
none of which is zero, the product of their least common 
multiple and the greatest common factor of the n prod- 
ucts, each having as factors all but one of the given 
integers, 1s equal to the numerical value of the product 
of the given integers (’). 

This follows from § 719 by supposing 


n° 


y= A, 4,4.° °° 


(1) See Barrieu, p. 217; Lucas, p. 345; Tannery, p. 124. 
(2) See Lucas, p. 346; Barrieu, p. 217; Stieltjes, p. 2. 


238 INTEGERS. 


723, Theorem. //aving given a series of n integers, 
none of which ts zero, the product of the least common 
multiple of all the products that can be forma, each hav- 
ing as factors k of these integers, wher k <n, and the 
greatest common factor of all the products that can be 
Jormd, each having as factors n — k of these integers, 
ws equal to the numerical value of the product of the 
given integers ('). 


Vet, 0.) a, 4 2 sbemtneareivenintesersm and 


y= GA, 4,4: °° 


n* 


Let also f), 83; 8, --:, 2, be the senesiof products 
of the a’s taken & at a time. 

Then » is a common multiple of the f’s and 
VR, Vi Vitae) ads the scries:otaprod ker 
of the a’s taken z — & at a time. 

Now 


(3, XP, X Bs X--- X B,) 


x [YAO PRS) CONN |». 
719. 


Hence the theorem is proved. 

This theorem contains the preceding as a particu- 
lar case. 

724, Theorem. Jf v 7s a common multiple of the in- 
legers Gy, A,, Ay +++, a, then (4,@Q4,Wa,&---&a4,) 
[ r4) X@i4)X Via) xX Xera)l=[y], 


provided v 1s not Zero, 





— 


(1) See’ Lucas: p. 346; Barrie, p..217. 


a 


LEAST COMMON MULTIPLE. 239 


This theorem is proved from § 699 by the method 
by which § 719 was proved from § 697. 

725. Theorem. Having given a series of n integers, 
none of which ts zero, the product of their greatest com- 
mon factor and the least common multiple of the n prod- 
ucts, each having as factors all but one of the given in- 
tegers, 1s equal to the numerical value of the product of 
the given integers ('). 

This follows from § 724 by supposing 

y= GZ, A,4, se a 

Or this theorem may be proved from § 723 by sup- 
posing the & of that theorem equal to z — 1 and using 
the commutativ law for multiplication. 

726. Theorem. Having given a series of n integers, 
none of which rs zero, the product of the greatest common 
factor of all the products that can be formd, each hav- 
ing as factors k of these integers, wher k <n, and the 
least common multiple of all the products that can be 
formd, each having as factors n — k of these integers, 
7s equal to the numerical value of the product of the 
given integers (*). 

This theorem may be proved in the same way as 
§ 723. 

Or it follows from § 723 by putting z — & for £ and 
using the commutativ law for multiplication. 


(1) See Barrieu, p. 218 ; Stieltjes, p. 2. 
(2) See Barrieu, p. 219. 


240 INTEGERS. 


727, Theorem. J// the integers a,, G,, Gs, +++, none of 
which ts zero, ar prime to each other two by two, their 
least common multiple 1s | a,a,4,-+-|, and conversely ('). 

For the direct theorem, setting v= |4a,a,a,.--], 
since @,, 4, 4, +++ ar prime to each other two by two, 
(Bos a0 (Vit 2.) 00 (Yea Osa § 660. 

Therfor » is the least common multiple of a, a,, 
hg oO S20, 

For the converse theorem, since » is the least 
common multiple of @,, ¢,, 4 


ORE EES SS 
(¥: 4) QO: 4) WY: 4)®Q---= 1. § 720. 
Therfor @,, @,, @,, --- ar prime to each other two by 
two. § 666. 


728, Theorem. Jf che integers 0,, ,, a, +++, none of 
which is sero, ar prime to each other two by two, and 0 
7s any integer not zero, the least common multiple of 


Oa,, Oa,, Oa,, --- 7s | Aa,a,0,---|, and conversely. 
This follows from the preceding theorem, since 
(G4) X (8a,) X (G45) X +--+ =|G|(4 Xa Xa, X---), 


SWAle 
729, Theorem. Jf pu zs the positiv integer which has 
every prime factor that occurs tn any one of a series of 
positiv integers, and which contains each prime factor a 
number of times equal to the greatest number of times 
that it occurs as a factor in the given integers, then pts 
the least common multiple of those integers. 
First, since m contains every prime factor that 
occurs in any one of the given integers a number of 





(1) See Tannery, p. 121. 


ee 


LEAST COMMON MULTIPLE. 241 


times not less than the number of times that it occurs 
in that integer, “is a common multiple of the given 
integers. § 640. 

Now let » be any positiv common multiple of the 
given integers. 

Then every prime factor of any one of the given 
integers must be a factor of » and must occur in v a 
number of times not less than the number of times 
that it occurs in any one of the given integers (§ 640), 
and hence not less than the greatest number of times 
that it occurs as a factor in the given integers. 

Therfor every prime factor of “ occurs in vy a num- 
ber of times not less than the number of times that it 
occurs in #4. 

Hence » is a multiple of yu. 

Since, therfor, #7 is a common multiple of the given 
integers and every positiv common multiple is a mul- 
tiple of yw, # is the least common multiple of the given 
integers, 


16 


Cilia PE Rees 
CONGRUENCE. 


730. If ~ is any given integer, # and ¢ any integers, 
and we seta=f+ oy, thna—fB=gyp. By giving 
8 and ¢ various values we may thus obtain any num- 
ber of pairs of integers which differ by a multiple of 
pw. This is the justification for the existence of the 
following definition. 

731. Definition. An integer a is congruent (') with 
an integer $ with respect to an integer y, calld the 
modulus of congruence, when the difference a — Pf isa 
multiple of y. 

Thus, if a — 8 = gp, @ is congruent with # with re- 
spect to the modulus y. 

The statement of congruence is written 


a. = 2 (mod. p), 


which is read “a is congruent with 8, modulo yp.” 
The statement 4 = (mod. yp) is calld a congru- 
ence; « is its left member and { its right. 
When ther is no doubt what the modulus is, or 
when several congruences ar supposed to hold with 





(1) The idea of congruence is due to Gauss. See his ‘‘ Disquisitiones 
Arithmetice,’’ p. 1. See also Dirichlet-Dedekind, p. 33; Chrystal, 
‘¢Text-Book of Algebra,’’ Part II., p. 500; Tannery, p. 454. 


242 


+ 


CONGRUENCE. 243 


respect to the same modulus, we may write simply 
a= fp. 

If a is not congruent with § with respect to a given 
modulus 4, we may write a + 8 (mod. p). 

732. If a=f (mod. 0), a—f=¢-o0=0 and 
i—PeeanGe conversely, if a= 8 4—B=o= @-0 
and a = # (mod. 0). | 

That is, if the modulus were zero, the congruence 
a = would be equivalent to the equality a=. 
The idea of congruence would therfor, with a zero 
modulus, be equivalent to the old idea of equality. 
It would be nothing new. 

We will, therfor, in the future assume that the 
modulus is not zero. 

733. Theorem. <Axy integer 1s congruent with wself 
with respect to any modulus, 

Fora—a=o-p. 

734, Theorem. /f/ a=, then a= 8 with respect to 
any modulus. 

735, Theorem. 

Tf «= B (mod. p), then 8 = a (mod. p). 

For, ifa—B=gp, P—a=(—o¢)p. 

736, Theorem. 

If a 8 (mod. pp), then B 32 a (mod. p). 

737. Theorem. <Axy two integers ar congruent with 
respect to unity as a modulus. 

For a — 8 =(a—f)x 1. 

738, Theorem. 

Tf a= B (mod. p), then o = B (mod. 1). 


244 INTEGERS. 


For, ifa—B=gp, 4—B=(— 9). 
739. Theorem. 
If «= B (mod. p), then a = B (mod. p). 

740. Theorem. /f a= (mod. p) and p=», then 
a = 8 (mod. v). 

For, ifa—S=guandu=yv, a—Pp=o. 

741. Theorem. /f a= £ (mod. p) and 

8 =7 (mod. p), then a = jz (mod. 4). 


Since a = f, a—Bp= op. 

SinceeP=y7, B-7=yp. 

Adding, od fi amy aot 

Therfor aA=y7. 

742, Theorem. /faz= fi and 8 +7, thena +7. 

743. Theorem. // we hav a series of congruences in 
which the right member of each congruence, except the 
last, 1s also the left member of the succeeding congru- 
ence, ASaOSPSyped=Hec=..., then any integer in 
the series 1s congruent with any other integer in the 


SevteS. 

744, Theorem. [fa=f, thna+y=8+y7, and 
conversely. 

For the direct theorem, 
since a = f, a— p= on. 

Hence (4+ 7)—(P +7) = ¢p. 

Therfor a+y=P4+y;. 


For the converse theorem reverse the steps. 


CONGRUENCE. 245 


745. Theorem. Jf a=, then; +a=7+Ff, and 
conversely. 

746. Theorem. /f a=f8 and y=d, then 
a+y= P40, and conversely, if a+ty=P4+0 and 
yr =0, tena= fp. 

747, Theorem. Zhe sum of the left members of any 
number of congruences ts congruent with the sum of the 
right members. 

748. Theorem. /f a= f, then a—y=fP—yjy, and 
conversely, 

749. Theorem. /f a= f, theny —a=7— Ff, and 
conversely. 

750. Theorem. Jf a=fP and 7+ = 0, then 
a—y=Pp—od, and conversely, if a—y =P — 0 and 
y,=0,thna=fP; fa—y=P—danda= Ff, then 
y=. 

751. Theorem. /fa= P, then ay = fy. 


Since a4 = f, a— B= op. 


Hence (2 — B)r = (gp)p. 
Or ay — Br = (eye. 
Therfor ay = fy. 


752. Theorem. /fa= f, then ya = 7f. 

753. Theorem. /f a= f and 7 = 0, then of = Bo. 

754, Theorem. The product of the left members of 
any number of congruences ts congruent with the product 
of the right members. 


246 INTEGERS. 


755. Theorem. /f a= 8 (mod. p2), then 
ay = Br (mod, pr), provided 7 + 0; and conversely, if 
ay = Py (mod. pr), then a = 8 (mod. p). 

For the direct theorem, 
since 4 = # (mod. yp), a— B= op. 


Hence (a — B)r = (ep). 
Or ar — Br = 9(r7/). 
Therfor ay = fy (mod. pr). 


For the converse theorem reverse the steps. 

756. Theorem. /f a= 8 (mod. p) and p>y¥, then 
a= 8 (mod. »:7). 

Since a = 8 (mod. p), a—f=ogyp. 

Sincegs Sane) ay Ae ae 


Hence a— 3 = (gr). 
Therfor a = #8 (mod. »). 
Or a = 8 (mod. p#: 7). 


This theorem may be stated: /f two integers ar 
congruent with respect to a given modulus, they ar also 
congruent with respect to any factor of that modulus as 
a modulus. 

757. Theorem. /f a= (mod. p) and a, 8, and p 
ar all divisible by 7, thena:y=P:7 (mod. p:7), and 
conversely. 

Sifice 9 a.) Py. and» 2 Sar alle divisible by aay 
a= 07,8 = ey, and p = ry. 

Hence, since a4 = § (mod. yp), 


+? 


CONGRUENCE. 247 
Oy = ey (mod. v7). 
Therfor 0 = ¢ (mod. »). $9755: 
choc (Guerayal 0 Fay 


For the converse theorem reverse the steps. 


Or Cai 


758. Theorem. //f a= (mod. p) and a and f ar 
both divisible by y, thena:y7 = 8:7 (mod. wp: (U®&7)). 

Since a = ? (mod. p), 4 — 8 = op. 

ete7i00 7 .— «. 

Then # = v« and 7 = ox, wher »v || 0. § 604. 

Also a — 8 = gyn. 

Sincesas> 7 and -6.> 7, a— 6:>> 7 and 


(w—A):y—a:7—Pir. ‘8.479. 
Since a— >>, PE > OK 
and (2 — 2): 7 = (gre) : (Ox). § 460. 
Hence gy >> 6 and (a — f):7 = (gr) : 0. 
Since gy >>d andy] od, o>. S28, 
Hence (a — 8) :7=(¢: OV. § 467. 


ieee a9 3 yi (P's 0)( 2.3: 4). 
Therfor a:7=Ph:7 (mod. pw: x). 
Or Gps 7 (mod. fs ( (4607) ). 


Ifesy, p@r=\|rl- 

Hence, in this case, a: y = 8: 7 (mod. #: |7)). 

And Gp 7 (mods pity). 

This theorem, therfor, when 4 >>+7, reduces to the 
direct part of § 757. 


248 INTEGERS. 


759, Theorem. 

If a = B (mod. p) and a and 8 ar both divisible by x, 
wher p \\ y, thena:7=P:7 (mod. p)(?). 

Mee Mahe y Veit: 

760. Theorem. <Axzy tnteger 1s congruent with each 
of the remainders that it leaves when divided by the 
modulus. 

For, ifa=ou+p, a—p=oy. 

761. Theorem. /f «=p (mod. »), wher |p| <|p|, 
then p is one of the remainders obtaind by dividing a 
by pL. | § 521. 

762. Theorem. /f two integers ar congruent, they 
leave the same remainder when divided by the modulus, 
and conversely. 

For, if 8 leaves the remainder p, $= p. 

Hence, ifa=f8, a=p, wher |p|<|z|. 

Therfor, when @ is divided by y, a remainder is p. 

| § 761. 

Conversely, if a and # both leave the remainder p, 
a=pand P= op. 

Therfor a = f. 

763. Theorem. // two integers ar congruent and 
either ts atvisible by the modulus, the other ts also; and 
conversely, if each of two integers is divisible by the 
modulus, they ar congruent. 

764. Theorem. /f two integers ar congruent and 
either is divisible by a factor of the modulus, the other ts 
also. 





(1) See Tannery, p. 466. 


CONGRUENCE, 249 


765. Theorem. /f pw 7s any given modulus, all the 
integers, when written in the order of increasing mag- 
nitude, may be divided into successiv groups of | | each, 
so that the integers in cach of these groups, taken in 
order, ar congruent with O, 1, 2, 3, +++, |f#|— 1. 

The first integers in the successiv groups will be 
the various multiples of | 
==) — 34, — 2f, — #9; [5 2H, 3H o> 

if 4 is positiv, 
im aeons O, a fy at, — Sfty 
if # is negativ, 

If gy is either of these multiples, the group of 
which it is the first integer will be 

GH, PATI, PU + 2, GE + 3, °° oe + (le|— I). 

These integers ar in each case evidently congruent 
with 0, I, 2, 3, ---,|#|—TI. 

766. Definition. This principle is calld the periodicity 
of the integers with respect to a given modulus. 

767. Definition. A cyclical arrangement of any set 
of things, given in a certain order, as, for example, 
the letters a, 0, c,d, e, f, is an arrangement obtaind 
by dividing the set into two parts at any place 
and then making the first part the second and vice 
versa, the order of the symbols in each part being 
unchanged. 

fulmistU cra, ¢,}..0 ande, 7, 2, 0,.c) da ar cyclicalar- 
rangements of the above set of letters. 

The original set unchanged is also said to be a 
cyclical arrangement of itself. 


250 INTEGERS. 


768. Theorem. // ps 7s any given modulus and any 
group of | | successtv integers ar taken, they ar con- 
gruent, in order, with some cyclical arrangement of the 
mtegers O, 1, 2, 3, ++ |#|— 2, |#|—1. 

Let gyu+p be the first of the given group of 
integers, wher p is positiv and numerically less than yp. 

S522; 

Then the group of integers will be 


PEE TICs PESTO ak POM Tene teases 
get p+ (|2|—2), put p+ (|#|— 1): 
These differ each by a multiple of u from the cor. 
responding integers of the group 
PrP +1, 9 +2,-+,P+(|H|— 2) p+ Ue] — 2). 
Therfor the integers of the first group ar respectivly 
congruent with these integers. 
Now set | “| = + «, wher « is positiv. 
. Then the latter group may be written 
Ps Ber eee “+4 0 + (K — 2), pt(«e*—1) pt, 
pt(«et+ 1); Sera at i (l#|—2), e+ (/e|— 1), 
or 
C0 = I, ce Sete a || — 2, luz|— I, le], le | + 1 
Se Fe tO eeee | Hl aren) 
These ar congruent respectivly with 
P; p+ I, pP+2, oa) Iv|—2, l4l—1, ENS Seva P—2, p—I. 


Therfor the original group of integers ar congruent 
respectivly with these integers, which ar a cyclical ar- 


CONGRUENCE. 251 


rangement of the integers 
O, I, 2, 3,--,|4|—2,|#|—1. 


769. Theorem. Any group of |p| successtv inte- 
gers ar congruent, in order, with respect to the modu- 
lus pt, with some cyclical arrangement of the set 
of |p| ttegers obtaind from the set of integers 
0, I, 2,3,--,|/4#/—3,|#|— 2, |“#|—1 oy replacing 
any or all of them by the corresponding integers of the set 
eee el) VIE |— 2); — (Al. 3) os 

—3,—2,—1. 

For, if @ is the positiv remainder obtaind by ap- 
proximately dividing a given integer by yp, and o the 
corresponding negativ remainder, ¢ = — (| #| — p). 

§ 518. 

770, Theorem. In particular, any group of | p| suc- 
cessiv integers ar congruent, in order, with the integers 


—a, —(4—1), —(@— 2), --., 
ae om 2 are by O, 7), 2; Se ok, 


|u| —a—2,|v|—a—1, 


wher @ ts either zero or positiv and less than | p\. 


FUNDAMENTAL IDEAS. 


1. Number. Soe 
2. Equality. § 5. 
Seoul: 337. 

AXIOMS. 
I. a@=a4. § 6. 
2. Ifa= 4, then d= a. So73 
3. If@=6 and d=c, then a=e., § 34. 
4. @atb=db+a. Soa 
5. Ifa=J, thna+c- J. § 43. 
6. If a + 4, ther is some number c¢, such that either 
a=ct+borc+ta=d. § 45. 
7, Ifa=4, thena¢c=b+. § 46. 
8. Ifa+,thena+c+db+e. § 47. 
9g. (a+ 6)+c=a4(b4 0). § 54. 
10. Every number, except one, is the sum of two 
numbers. § OI. 


The author does not include § 40 in this list of axioms, as he has 
become convinced, since the electrotyping of most of the book that § 40 
is not properly an axiom. That the number of objects in the group ob- 
taind by joining two groups should be unique is a prerequisit to the 
definition of the symbol a + 6. If this number were not one-valued, 
a + 6 would not be defined. See § 416. 


202 


Bia eolreAUTHORSZAND BOOKS -REFERD 
TOR Ne ei CMI NG PAGES: 


ARCHIMEDES. 

W. W. R. Batt, A Short Account of the History of Mathematics, 1893. 

P. BARRIEU, Mathesis, 1883. 

BEMAN AND SMITH, Elements of Algebra, 1900 ; New Plane and Solid 
Geometry, I901. 


BHASKARA. 

G. CANTOR, Sur les Fondements de la Théorie des Ensembles Trans- 
finis, tr. from the German by F. Marotte, 1899. 

fig, CAYLEY. 

G. CHRYSTAL, Text-Book of Algebra, 1893. 

A. C. CLAIRAUT. 

L. CREMONA, Elements of Projective Geometry, tr. from the Italian 
by C. Leudesdorf, 1885. 

E. W. Davis, Logic of Algebra, 1890. 

R. DESCARTES, La Géométrie, 1637. 

U. Dini, Grundlagen fiir eine Theorie der Functionen einer verander- 
lichen reelen Grésse, tr. from the Italian by Liiroth and Schepp, 1892. 

DIRICHLET-DEDEKIND, Vorlesungen iiber Zahlentheorie, 4th ed., 1894. 


>» 


ERATOSTHENES. 

Euc ip, Elements of Geometry. 

L. EuLer, Elements d’ Algébre, ed. of 1807. 

FISHER AND SCHWATT, Text-Book of Algebra, Part I., 1898. 

K. F. Gauss, Disquisitiones Arithmeticze, 1801. 

J. WitLArp Gisps, Vector Analysis, 1884. 

A. GIRARD, Invention nouvelle en l’algébre, 1629. 

D. F. Grecory, On a Difficulty in the Theory of Algebra, 1840. 

J. HADAMARD, Lecons de Geométrie élémentaire, 1898. 

Wm. RowAn HAMILTON, Transactions of the Royal Irish Academy, 
Vol. XVII., 1835 ; Taylor’s Philosophical Magazine, 1844; Lectures 
on Quaternions, 1853. 


253 


254 LIST OF AUTHORS AND BOOKS. 


H. HANKEL, Theorie der komplexen Zahlsysteme, 1867. 

T. Harriot, Artis Analyticze Praxis, 1631. 

D. HILBERT, The Foundations of Geometry, tr. from the German by 
E. J. Townsend, 1902 ; Bulletin of the American Mathematical So- 
ciety, Vol. VIII., 1902. 


FE. V. HunTINGTON, Transactions of the American Mathematical So- 


ciety, Vol. III., 1902. 

Jevons-HILL, Elements of Logic, 1883. 

V. A. LEBEsGUE, Théorie des Nombres, 1862. 

A. LEFEVRE, Number and its Algebra, 1896. 

A. M. LEGENDRE, Théorie des Nombres, 4th ed., 1900. 

LEONARDO of Pisa, Liber Abaci, 1202. 

J. Locker, An Essay Concerning Human Understanding, 1690. 

E. Lucas, Théorie des Nombres, 1891. 

E. H. Moore, Transactions of the American Mathematical Society, 
1902. 

JorDANUS NEmMoRARIUS, Algorithmus Demonstratus, 1534. 

W. OUGHTRED, Clavis Mathematica Denuo Limata, 1648. 

G. Peacock, Symbolical Algebra, 1845. 

R. M. Pierce, Problems of Number and Measure, 1808. 

R. REcorpD, Whetstone of Witte, 1557. 

FE. ScHRODER, Abriss der Arithmetik und Algebra, 1874. 

H. ScHuBERT, Arithmetik und Algebra, 1899; Encyklopadie der 
Mathematischen Wissenschaften, Vol. I., 1898. 

J. A. SERRET, Traité d’ Arithmétique, 1852. 

J. F. Servos, Gergonne’s Annales, Vol. V., 1814. 

T. J. SrreLtTyes, Annales de la Faculté des Sciences de Toulouse, Vol. 
IV., 1890. 

M. STIFEL, Arithmetica Integra, 1544. 

STOLZ UND GMEINER, Theoretische Arithmetik, 1900. 

W. Swinton, New English Grammar, 1881. 

J. TANNERY, Lecons d’ Arithmétique, 1900. 

F. VieTA, In Artem Analyticam Isagoge, 1591. 

K. WEIERSTRASS, Gesammelte Werke, Vol. i 

G. A. WENTWORTH, Plane and Solid Geometry, I901. 

J. WrpmAn, Mercantile Arithmetic, 1489. 


+? 


INDEX. 


[The references ar to sections. ] 


Absolute value, 393 
Activ element, 42 
Addition, 42 
table, 74 
Affect, 257 
Algebraic sum, 224 
Algorithm of Euclid, 594 
Analytic combination, 150, 511 
operation, 150, 511 
Approximate division, 513 
Associativ law for addition, 54 
and subtraction of 
integers, 257 
of integers, 249 
greatest common factor, 


646 
least common multiple, 
708 
multiplication, 380 
Augend, 42 


Axiom, 6, 10 
of Archimedes, 504 


Base of system of numbers, 64 
product, 296 


Cancellation, 261 
Categorical statement, 12 
Cipher, 181 

Closed system, 295 








Coefficient, 296 
Common factor, 551 
multiple, 674 
Commutativ law for addition, 41 
and subtraction of 
integers, 259 
of integers, 207 
equality, 7 
of integers, 171 
greatest common factor, 


574 


least common multiple, 
681 
multiplication, 375 
Compatible, 10 
Complex algebraic sum, 247 
product, 378 
sum, 51 
Composit integer, 538 
Compound categorical statement, 18 
conditional statement, 20 
statement, 18 
Conclusion, 12, 130 
Conditional statement, 12 
Congruence, 731 
Consistent, Io 
Constructiv syllogism, 131 
Contradictory statement, 13 
of compound categorical 
statement, 19 


255 


256 INDEX. 


Converse of hypothetical statement, ; Exact division, 513 


pas Explicit definition, 84 
Correspondence, 70 
Counting, 71 Factor, 376 
Cyclical arrangement, 767 Of.427 


Finite number, 137 . 
Derived idea, 84 

ES SEN syllogism, 32 Generalized associativ and commu- 
Determinate expression, 39 


tativ law, for 
Difference, 104 


addition, 59 


of integers, 220 
of integers, 250 


Direct operation, 150, 511 


proof, 132 greatest common 


factor, 647 
least common mul- 
tiple, 709 
multiplication, 381 
Generalized distributiv law for di- 


statement, 21 
Disjunctiv statement, 12 
syllogism, 131 
Distributiv law for division and ad- 
dition, 478 
greatest common 
factor, 602 
least common 
multiple, 704 


vision and greatest 
common factor, 
656 


least common multi- 


Dividend, 506, 513 pleas 
Divisibili left-handed distributiv law for 
ptt sete multiplication 


Division, 506, 513 and addition 


table, 508 Fe 
Divisor, 506, 513 309 
of, 409 greatest common 


factor, 651 
least common 


Duality, 694 


Elements of difference, 77 multiple, 713 
greatest common factor, 572 right-handed distributiv law for 
integer, 155 multiplication 
least common multiple, 679 and _ addition, 
sum, 37 370 

Equality, 5 greatest common 
of integers, 168 ; factor, 652 

Even integers, 287 least common 


numbers, 68 multiple, 714 


INDEX. 257 


Greater, 75 

in integers, 186 
Greatest common factor, 572 
Group, 3 


Hypothesis, 12 
Hypothetical statement, 12 


Implicit definition, 84 

Increment, 42 

Independent axioms, 10 

Indeterminate symbols, 416 

Indirect operation, 150, 511 
proof, 132 

Inequality, 5, 75 

Infinit number, 137 

Integer, 153 

Inverse, 147 


Larger than, 91 
Least common multiple, 676 
Left-handed difference, 93 
distributiv law for multiplica- 
tion and ad- 
dition, 310 


greatest com- 
mon factor, 
598 

least common 


multiple, 701 
quotient, 429 
ratio, 433 
remainder, 97 
BESS, °75 
in integers, 186 
Lie between, 140, 285 
Lower quotient, 513 


Mathematical induction, 607 
a7 








Members of congruence, 731 
equality, 5 

Method of exclusion, 132 
exhaustion, 132 

Minuend, 145 

Modulus of congruence, 731 
integer, 393 

Multiple, 296 

Of; 427 

Multiplicand, 297 

Multiplication, 297 

table, 392 

Multiplier, 297 

Multivalent symbols, 416 


Names for numbers, 65 
Natural series of integers, 280 
numbers, 67 
Naught, 181 
Necessary condition, 127 
Negativ integers, 159 

of integer, 161 
Number, I 

couple, 153 
Numerical value, 393 





Odd integers, 287 
numbers, 68 
One, 2 
Operand, 42 
Operation, 42 
of finding greatest common fac- 


tor, 573 
least common multiple, 
680 
Operations of the first degree, 
511 


second degree, 511 


258 INDEX, 


Opposit of conditional statement, 16 


integer, 161 
Order, 70 

of magnitude, 144 

of integers, 279 
Parenthesis, 51 
Partial converse, 32 

products, 373 
Parts, 60 
Passiv element, 42 
Periodicity of integers, 766 
Positiv integers, 159 
Postfactor, 376 
Prefactor, 376 
Premis, 130 
Primary numbers, I51 
Prime integer, 538 

to, 555 

two by two, 662 

Product, 296 

coefficient an integer, 313 
Quotient, 440 
Reciprocity, 694 
Reductfon to absurdity, 132 
Remainder, in division, 514 

subtraction, 81 
Resolution into ones, 62 

prime factors, 550 

Right-handed difference, 77 


distributiv law for multiplica- 
tion and addi- 


tion, 311 


greatest common 


factor, 599 


least common 


multiple, 702 
quotient, 412 
ratio, 418 








Right-handed remainder, 81 
Rule of signs for division, 490 


multiplication, 


326 
Simple categorical statement, 18 
conditional statement, 20 
statement, 18 
sum, 51 
Siv of Eratosthenes, 545 
Smaller than, 91 
Standard algebraic sum, 258 
sum, 53 
Subscripts, 157 
Subtraction, 145 
table, 146 
Subtrahend, 145 
Subtrahent, 145 
Successiv addition, 51 
Sufficient condition, 127 
group of axioms, IO 
Sum, 37 
of integers, 203 
Summand, 42 
Syllogism, 130 
Symbols for numbers, 65 
Synthetic combination, 150, 511 
operation, 150, 511 
Table of prime numbers, 545 
Theorem, II 
Transposition, 262 
Unambiguous expression, 39 
Unconditional statement, 12 
Unit, 64 
Unity, 64 
Univalent symbol, 39 
Upper quotient, 513 
Zero, 181 
Zero integers, 159 


+? 











C001 


ELEMENTS OF THE THEORY OF INTEGERS NEW 


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